Variational Operators, Symplectic Operators, and the Cohomology of Scalar Evolution Equations

TL;DR

This paper investigates the cohomology of scalar evolution equations, revealing the structure of variational and symplectic operators, and characterizing third-order equations with specific symplectic properties.

Contribution

It establishes the isomorphism between cohomology spaces and variational or symplectic operators, and characterizes third-order equations with first-order symplectic operators.

Findings

Cohomology space H^{1,2} is isomorphic to variational and symplectic operators.

H^{1,s} vanishes for s≥3, simplifying the cohomology structure.

Characterization of third-order scalar evolution equations with first-order symplectic operators.

Abstract

For a scalar evolution equation $u_t=K(t,x,u,u_x,\ldots, u_n), n\geq 2$ the cohomology spaces $H^{1,s}({\mathcal R}^\infty)$ vanishes for $s\geq 3$ while the space $H^{1,2}({\mathcal R}^\infty)$ is isomorphic to the space of variational operators. The cohomology space $H^{1,2}({\mathcal R}^\infty)$ is also shown to be isomorphic to the space of symplectic operators for $u_t=K$ for which the equation is Hamiltonian. Third order scalar evolution equations admitting a first order symplectic (or variational) operator are characterized. The symplectic nature of the potential form of a bi-Hamiltonian evolution equation is also presented.

Equations

$$m\cdot \Delta =\mathbf{E}(L)$$

$$\mathcal{E}(\Delta )=\mathbf{E}(L)$$

$$t{D}_x\left(u_t-u_{xxx}-\frac{1}{2}{u}_x^2+\frac{u}{2t}\right)=\mathbf{E}\left(-\frac{1}{2}t{u}_x{u}_t+\frac{1}{2}t{u}_x{u}_{xxx}+\frac{1}{6}t{u}_x^3\right).$$

$$\omega =dx\wedge {\theta }^0\wedge ϵ-dt\wedge \sum _{j=1}^{2m+1}\left(\sum _{a=1}^j(-X{)}^{a-1}\left(\frac{\partial K}{\partial u_j}ϵ\right)\wedge {\theta }^{j-a}\right)$$

$$\Phi (\mathcal{E})=[\omega ],$$

$$\mathcal{S}(K)=\mathbf{E}(L).$$

$$\kappa ={\hat{K}}_{2}dx+\left(-2K_0+K_1{\hat{K}}_2-\frac{1}{2}\left(X(K_3){\hat{K}}_2^2+K_3{\hat{K}}_2^3\right)+X\left(K_{3}X({\hat{K}}_2)\right)\right)dt$$

$$u_t=\mathcal{D}\circ \mathbf{E}(H)$$

$$D_t$$

$$D_x$$

$${\vartheta }^0$$

$${\vartheta }^i$$

$${\zeta }^{a,j}$$

$$\omega =dt\wedge \alpha +dx\wedge \beta ,\alpha ,\beta \in \mathrm{span}\{{\vartheta }^i\wedge {\vartheta }^j,{\vartheta }^i\wedge {\zeta }^{a,j},{\zeta }^{a,i}\wedge {\zeta }^{b,j}\}.$$

$$d_{\text{scaleto}H4pt}:{\Omega }^{r,s}(J^{\infty }(\mathbb{R},\mathbb{R}))\to {\Omega }^{r+1,s}(J^{\infty }(\mathbb{R},\mathbb{R})),d_{\text{scaleto}V4pt}:{\Omega }^{r,s}(J^{\infty }(\mathbb{R},\mathbb{R}))\to {\Omega }^{r,s+1}(J^{\infty }(\mathbb{R},\mathbb{R}))$$

$$d_{\text{scaleto}H4pt}f=D_t(f)dt+D_x(f)dx,d_{\text{scaleto}V4pt}f=\frac{\partial f}{\partial u_i}{\vartheta }^i+\frac{\partial f}{\partial u_{t,i}}{\zeta }^{1,i}+\dots ,$$

$$d_{\text{scaleto}H4pt}{\vartheta }^i=-{\zeta }^{1,i}\wedge dt-{\vartheta }^{i+1}\wedge dx,d_{\text{scaleto}V4pt}{\vartheta }^I=0,$$

$$d_{\text{scaleto}H4pt}^2=0,d_{\text{scaleto}V4pt}^2=0,\mathrm{and}{d}_{\text{scaleto}H4pt}{d}_{\text{scaleto}V4pt}+d_{\text{scaleto}V4pt}{d}_{\text{scaleto}H4pt}=0.$$

$$I(\omega )=\frac{1}{s}{\vartheta }^0\wedge \sum _{a=0,i=0}^{\infty }(-1{)}^{i+a}{D}_x^i{D}_t^a(({\partial }_{u_{a,i}}\text{raise}0.5pt\text{vbox}\text{hrule}height=0.4pt,width=6.0pt,depth=0.0pt\text{vrule}height=6.0pt,width=0.4pt,depth=0.0pt\omega ),$$

$$I^2=I,\omega =I(\omega )+d_{\text{scaleto}H4pt}\eta ,\mathrm{for}\mathrm{some}\eta \in {\Omega }^{1,s}(J^{\infty }({\mathbb{R}}^2,\mathbb{R})).$$

$$J(\kappa )=\sum _{a,i=0}^{\infty }(-1{)}^{i+a}{D}_x^i{D}_t^a(({\partial }_{u_{a,i}}\text{raise}0.5pt\text{vbox}\text{hrule}height=0.4pt,width=6.0pt,depth=0.0pt\text{vrule}height=6.0pt,width=0.4pt,depth=0.0pt\kappa )$$

$$\ker J=\ker I=\mathrm{Im}{d}_{\text{scaleto}H4pt}.$$

$$(\rho \wedge \mathcal{E}(\omega )-{\mathcal{E}}^{\ast }(\rho )\wedge \omega )\wedge dt\wedge dx=d_H\tilde{\zeta }.$$

$${\mathcal{E}}^{\ast }(\alpha )=(-1{)}^{i+a}{D}_x^i{D}_t^a(r_{ai}\alpha ),\alpha \in {\Omega }^{r,s}(J^{\infty }({\mathbb{R}}^2,\mathbb{R})).$$

$$\Delta =u_t-K(t,x,u,u_x,\dots ,u_n),$$

$$d_{\text{scaleto}V4pt}\Delta ={\vartheta }_t-K_i{\vartheta }^i\mathrm{where}{K}_i=\frac{\partial }{\partial u_i}K(t,x,u,u_x,\dots ,u_n),i=0,\dots ,n.$$

$${\mathbf{F}}_{\Delta }=D_t-\sum _{i=0}^n{K}_i{D}_x^i.$$

$${\mathbf{F}}_{\Delta }^{\ast }(\rho )=-D_t(\rho )-\sum _{i=0}^n(-D_x{)}^i(K_i\rho ),\rho \in {\Omega }^{\ast }(J^{\infty }({\mathbb{R}}^2,\mathbb{R}).$$

$$\iota =[t=t,x=x,u=u,u_t=K,u_x=u_x,u_{tt}=T(K),u_{tx}=X(K),u_{xx}=u_{xx},\dots ],$$

$$X$$

$$T$$

$${\theta }^i={\iota }^{\ast }{\vartheta }^i=d{u}_i-X^i(K)dt-u_{i+1}dx.$$

$$\{dt,dx,{\theta }^i=d{u}^i-X^i(K)dt-u_{i+1}dx\}i=0,1,\dots$$

Full text

1. Introduction

Given a scalar differential equation $\Delta=0$, the multiplier problem in the calculus of variations consists in determining whether there exists a smooth function $m$ (the multiplier) and a smooth Lagrangian $L$ such that

[TABLE]

where ${\bf E}$ is the Euler-Lagrange operator and ${\bf E}(L)$ is the Euler-Lagrange expression for $L$. The problem of determining whether $m$ and $L$ exists has a long history and is known as the inverse problem in the calculus of variations [anderson-thompson:1992a, fels:1996a, Douglas:1941a, anderson-duchamp:1984a, saunders:2010a, do-prince:1941a, krupka:1981a].

The variational bicomplex [anderson:2016a, anderson:1992a, tsujishita:1982a] can be used to provide an invariant solution to the inverse problem by utilizing the Helmholtz conditions. In terms of the variational bicomplex, the existence of a solution to 1.1 can be expressed by the existence of an element of the cohomology space $H^{n-1,2}$ for the equation of a special algebraic nature where $n$ is the number of independent variables. The existence of a non-trivial cohomology class for the equation $\Delta$ can then in principle be expressed in terms of invariants of the equation such as in [anderson-thompson:1992a, fels:1996a].

One of the goals of this article is to give a complete interpretation of the cohomology space $H^{1,2}({{\mathcal{R}}^{\infty}})$ for scalar evolution equations $u_{t}=K(t,x,u,u_{x},\ldots)$ which extends the interpretation of the special elements which control the solution to the inverse problem. The result is a natural generalization of the inverse problem in equation 1.1 we call the variational operator problem. Given a differential equation $\Delta=0$, does there exist a differential operator ${\mathcal{E}}$ and Lagrangian $L$ such that

[TABLE]

A simple example is given by the potential cylindrical KdV equation, $u_{t}=u_{xxx}+\frac{1}{2}u_{x}^{2}-\frac{u}{2t}$ which admits ${\mathcal{E}}=tD_{x}$ as a first order variational operator,

[TABLE]

The variational operator problem in equation 1.2 can be studied for either the case of scalar or systems of ordinary or partial differential equations. Here we restrict our attention to problem 1.2 in the case where $\Delta$ is a scalar evolution equations in order to relate this problem to the theory of symplectic and Hamiltonian operators for integrable systems.

In Section 2 we give a quick summary of the relevant facts about the variational bicomplex for the case we need. Sections 3 and 4 provide normal forms for the cohomology spaces $H^{r,s}({{\mathcal{R}}^{\infty}})$ in the variational bicomplex associated with the equation $\Delta=0$. These forms are then used in Section 5 to show there exists a one to one correspondence between the solution to 1.2 and the cohomology space $H^{1,2}({{\mathcal{R}}^{\infty}})$. Even order evolution equations don’t admit variational operators but we have the following theorem for odd order equations (the summation convention is assumed).

Theorem 1.1**.**

Let ${\mathcal{E}}=r_{i}(t,x,u,u_{x},\ldots)D_{x}^{i}$ $i=0,\ldots,k$ be a $k^{th}$ order differential operator and let the zero set of $\Delta=u_{t}-K(t,x,u,u_{x},\ldots,u_{2m+1}),m\geq 1$ define an odd order evolution equation.

1. (1)

The operator ${\mathcal{E}}$ is a variational operator for $\Delta$ if and only if ${\mathcal{E}}$ is skew-adjoint and

[TABLE]

is $d_{H}$ closed on ${{\mathcal{R}}^{\infty}}$, where $\epsilon=-\frac{1}{2}r_{i}\theta^{i}$ and $\theta^{i}$ are given in equation 2.11. 2. (2)

Let ${\mathcal{V}}_{op}(\Delta)$ be the vector space of variational operators for $\Delta$. The function $\Phi:{\mathcal{V}}_{op}(\Delta)\to H^{1,2}({{\mathcal{R}}^{\infty}})$ defined from equation 1.3 by

[TABLE]

is an isomorphism.

The isomorphism property of $\Phi$ in Theorem 1.1 implies that a scalar evolution equation admits a variational operator if and only if $H^{1,2}({{\mathcal{R}}^{\infty}})\neq 0$. Moreover the operator ${\mathcal{E}}$ (and subsequently the function $L$) in 1.2 are easily determined from $[\omega]\in H^{1,2}({{\mathcal{R}}^{\infty}})$ see Theorem 5.4.

Theorem 1.1 converts the solution to the operator problem 1.2 for a differential equation into a cohomology computation for the equation. The techniques developed for solving the multiplier inverse problem in terms of cohomology [anderson-thompson:1992a, fels:1996a, anderson-duchamp:1984a] can then be used to solve the operator problem.

A related problem to 1.2 in the theory of integrable systems is the notion of a symplectic Hamiltonian evolution equation [dorfman:1993a] which is reviewed in Section 6 in terms of the variational bicomplex. In the time independent case, a scalar evolution $u_{t}=K(x,u,u_{x},\ldots,u_{n})$ equation is said to be Hamiltonian with respect to a time independent symplectic operator ${\mathcal{S}}=s_{i}(x,u,u_{x},\ldots)D_{x}^{i}$ if

[TABLE]

For a time dependent equation and operator, condition 1.5 is given in 6.28 in terms of the symplectic potential. Symplectic operators exists on a different space than variational operators but there is a natural identification (see Remark 2.1) between symplectic operators and operators which can be variational operators. With this identification, problems 1.2 and 1.5 are shown to be the same and in Section 7 and we have the following theorem.

Theorem 1.2**.**

Let ${\mathcal{S}}=s_{i}(t,x,u,u_{x},\ldots)D_{x}^{i}$ be a differential operator and let $\Delta=u_{t}-K(t,x,u,u_{x},\ldots,u_{n})$. The operator ${\mathcal{S}}$ is a symplectic operator and $\Delta=0$ is a symplectic Hamiltonian system for ${\mathcal{S}}$ if and only if ${\mathcal{S}}$ is a variational operator for $\Delta$.

Theorem 1.2 shows that symplectic operators and variational operators for $u_{t}=K$ are the same so that Theorem 1.1 implies the following.

Theorem 1.3**.**

The function $\Phi$ in equation 1.4 defines an isomorphism between the vector space of symplectic operators ${\mathcal{S}}={\mathcal{E}}=r_{i}(t,x,u,u_{x},\ldots)D_{x}^{i}$ for which $\Delta=u_{t}-K$ is Hamiltonian, and the cohomology space $H^{1,2}({{\mathcal{R}}^{\infty}})$.

With Theorem 1.3 in hand, the determination of a symplectic Hamiltonian formulation of $u_{t}=K$ is resolvable in terms of the cohomology $H^{1,2}({{\mathcal{R}}^{\infty}})$ of the differential equation $u_{t}=K$ and subsequently the invariants of $\Delta$. This characterization of symplectic Hamiltonian evolution equations in terms of $H^{1,2}({{\mathcal{R}}^{\infty}})$ allows the techniques in [anderson-thompson:1992a, fels:1996a, anderson-duchamp:1984a] to be used in their study.

A key idea that directly explains the interplay between the symplectic Hamiltonian formulation for an evolution equation and the cohomology $H^{1,2}({{\mathcal{R}}^{\infty}})$ is the fact that the equation manifold ${{\mathcal{R}}^{\infty}}$ is canonically diffeomorphic to $\mathbb{R}\times J^{\infty}(\mathbb{R},\mathbb{R})$. The cohomology of the equation is expressed in terms of the geometric structure that arises from the embedding into $J^{\infty}(\mathbb{R}^{2},\mathbb{R})$ while the symplectic Hamiltonian formulation of an equation is expressed in terms of the contact structure on $\mathbb{R}\times J^{\infty}(\mathbb{R},\mathbb{R})$. Theorem 7.5 shows how these are related and this leads to Theorem 1.3.

In Section 8 the case of first order operators for third order equations are examined in detail and the following characterization is found.

Theorem 1.4**.**

A third order scalar evolution equation $u_{t}=K(t,x,u,u_{x},u_{xx},u_{xxx})$ admits a first order symplectic operator (or variational operator) ${\mathcal{E}}=2RD_{x}+D_{x}R$ if and only if $\kappa$ is a trivial conservation law, where

[TABLE]

and $K_{i}=\partial_{i}K$, $\hat{K}_{2}=\frac{2}{3K_{3}}(K_{2}-X(K_{3}))$, and $X$ is the total $x$ derivative on ${{\mathcal{R}}^{\infty}}$.

Furthermore, when $\kappa=d_{H}(\log R)$ then $u_{t}=K$ admits the first order symplectic (or variational) operator ${\mathcal{E}}=2RD_{x}+D_{x}R$

In Section 8 we examine the relationship between the Hamiltonian form of evolution equations and their potential form. In [Nutku:2002a] it is shown that the (first order) potential form of a Hamiltonian equation admits a variational operator. We examine this in more detail, as well as the role of bi-Hamiltonian systems as in [pavlov:2017a]. This theory is used in Example 9.1 where the Krichever-Novikov equation (or Schwartzian KdV) is shown to be the potential form of the Harry-Dym equation. The symplectic operators (or variational operators) for the Krichever-Novikov equation arise as the lift of the Hamiltonian operators of the Harry-Dym equation as described in Section 8.3.

Theorem 1.4 should be contrasted to the problem of determining a Hamiltonian formulation of a scalar evolution equation in terms of a Hamiltonian operator. An evolution equation $u_{t}=K$ is Hamiltonian with respect to a Hamiltonian operator ${\mathcal{D}}$ if there exists a Hamiltonian function $H$ (see [vino:1986a, dorfman:1993a, Olver:1993a]) such that

[TABLE]

Conditions for the existence of ${\mathcal{D}}$ and $H$ in equation 1.7 in terms of the invariants of $u_{t}=K$ is unknown. We illustrate the difference in these problems with the cylindrical KdV and its potential form. The potential form of the cylindrical KdV is easily shown to admit at least two variational (or symplectic) operators. Section 8.3 then suggests that the cylindrical KdV is a bi-Hamiltonian system. See Example 9.3 where a bi-Hamiltonian formulation of the cylindrical KdV is proposed ([wang:2002a] states that no Hamiltonian exists for the cylindrical KdV).

Lastly, in Appendix A we identify the elements of $H^{1,1}({{\mathcal{R}}^{\infty}})$ which don’t arise as the vertical differential of a conservation law with a family of variational operators. Example 9.2 demonstrates the theory.

2. Preliminaries

In this section we review some basic facts on the variational bicomplex associated with scalar evolution equations, see [anderson-kamran:1997a] for more details.

2.1. The Variational Bicomplex on $J^{\infty}(\mathbb{R}^{2},\mathbb{R})$

The $t$ and $x$ total derivative vector fields on $J^{\infty}(\mathbb{R}^{2},\mathbb{R})$ with coordinates $(t,x,u,u_{t},u_{x},u_{tt},u_{tx},u_{xx},\ldots)$ are given by

[TABLE]

The contact forms on $J^{\infty}(\mathbb{R}^{2},\mathbb{R})$ are

[TABLE]

where $u_{i}=D_{x}^{i}(u)$ and $u_{a,i}=D^{i}_{x}D^{a}_{t}(u)=u_{tttt\ldots,xxx\ldots}$.

The free variational bicomplex on ${J^{\infty}(\mathbb{R}^{2},\mathbb{R})}$ is denoted by $\Omega^{r,s}({J^{\infty}(\mathbb{R}^{2},\mathbb{R})})$ where $\omega\in\Omega^{r,s}({J^{\infty}(\mathbb{R}^{2},\mathbb{R})})$ is a differential form of degree $r+s$ which is horizontal of degree $r$ and vertical of degree $s$ (see Section 2 in [anderson-kamran:1997a]). In particular if $\omega\in\Omega^{1,2}({J^{\infty}(\mathbb{R}^{2},\mathbb{R})})$ then

[TABLE]

The horizontal and vertical differentials are anti-derivations

[TABLE]

which satisfy

[TABLE]

where $f\in C^{\infty}({J^{\infty}(\mathbb{R}^{2},\mathbb{R})})$. Since $d=d_{\scaleto{\mkern-1.5muH}{4pt}}+d_{\scaleto{\mkern-1.0muV}{4pt}}$ this implies,

[TABLE]

The integration by parts operator $I:\Omega^{2,s}(J^{\infty}(\mathbb{R}^{2},\mathbb{R}))\to\Omega^{2,s}(J^{\infty}(\mathbb{R}^{2},\mathbb{R}))$ is defined by

[TABLE]

and it has the following properties [anderson:1992a], [anderson:2016a],

[TABLE]

If we let $J:\Omega^{r,s}(J^{\infty}(\mathbb{R}^{2},\mathbb{R})\to\Omega^{r-1,s}(J^{\infty}(\mathbb{R}^{2},\mathbb{R}))$ be

[TABLE]

then $I(\kappa)=\frac{1}{s}\vartheta^{0}\wedge J(\kappa)$. Both $J$ and $I$ satisfy,

[TABLE]

The operator $J$ is the interior Euler operator, see page 292 in [anderson-kamran:1997a] or page 43 in [anderson:2016a].

Let ${\mathcal{E}}=r_{ia}D_{x}^{i}D_{t}^{a}$ be a total differential operator. The formal adjoint ${\mathcal{E}}^{*}$ is the total differential operator characterized as follows. For any $\rho\in\Omega^{0,s}({J^{\infty}(\mathbb{R}^{2},\mathbb{R})})$ and $\omega\in\Omega^{0,s^{\prime}}({J^{\infty}(\mathbb{R}^{2},\mathbb{R})})$ there exists $\zeta\in\Omega^{1,s+s^{\prime}}({J^{\infty}(\mathbb{R}^{2},\mathbb{R})})$ depending on $\rho$ and $\omega$ such that

[TABLE]

This leads to

[TABLE]

It follows from 2.6 that the formal adjoint satisfies $({\mathcal{E}}^{*})^{*}={\mathcal{E}}$.

Let $\Delta$ be a smooth function on $J^{\infty}(\mathbb{R}^{2},\mathbb{R})$. The Fréchet derivative of $\Delta$ [Olver:1993a] is the total differential operator ${\bf F}_{\Delta}$ satisfying $d_{\scaleto{\mkern-1.0muV}{4pt}}\Delta={\bf F}_{\Delta}(\vartheta^{0})$. If

[TABLE]

then

[TABLE]

The Fréchet derivative of $\Delta$ is determined from equation 2.7 to be the total differential operator

[TABLE]

The adjoint of the operator in 2.8 is,

[TABLE]

2.2. The Variational Bicomplex on ${{\mathcal{R}}^{\infty}}$ and $H^{r,s}({{\mathcal{R}}^{\infty}})$

Let $\Delta=u_{t}-K(t,x,u,u_{x},\ldots,u_{n})$ and let ${{\mathcal{R}}^{\infty}}$ be the infinite dimensional manifold which is the zero set of the prolongation of $\Delta=0$ in $J^{\infty}(\mathbb{R}^{2},\mathbb{R})$. With coordinates $(t,x,u,u_{x},u_{xx},\ldots)$ on ${{\mathcal{R}}^{\infty}}$ the embedding $\iota:{{\mathcal{R}}^{\infty}}\to J^{\infty}(\mathbb{R}^{2},\mathbb{R})$ is given by

[TABLE]

where $T$ and $X$ are the restriction of $D_{t}$ and $D_{x}$ to ${{\mathcal{R}}^{\infty}}$ given by,

[TABLE]

and satisfy $[X,T]=0$. The Pfaffian system ${\mathcal{I}}=\{\theta^{i}\}_{i\geq 0}$ on ${{\mathcal{R}}^{\infty}}$ is generated by the pullback of $\vartheta^{i}$ in equation 2.1

[TABLE]

The forms

[TABLE]

form a coframe on ${{\mathcal{R}}^{\infty}}$, and give rise to a vertical and horizontal splitting in the complex of differential forms leading to the bicomplex $\Omega^{r,s}({{\mathcal{R}}^{\infty}})$, $r=0,1,2$ and $s=0,1,\ldots$. For example if $\omega\in\Omega^{1,2}({{\mathcal{R}}^{\infty}})$ then

[TABLE]

where $\alpha=a_{ij}\theta^{i}\wedge\theta^{j}$ and $\beta=b_{ij}\theta^{i}\wedge\theta^{j}$, $a_{ij},b_{ij}\in C^{\infty}({{\mathcal{R}}^{\infty}})$. The bicomplex $\Omega^{r,s}({{\mathcal{R}}^{\infty}})$ is the pullback of the free bicomplex $\Omega^{r,s}({J^{\infty}(\mathbb{R}^{2},\mathbb{R})})$ by the embedding $\iota:{{\mathcal{R}}^{\infty}}\to{J^{\infty}(\mathbb{R}^{2},\mathbb{R})}$.

The horizontal exterior derivative $d_{\scaleto{\mkern-1.5muH}{4pt}}:\Omega^{r,s}({{\mathcal{R}}^{\infty}})\to\Omega^{r+1,s}({{\mathcal{R}}^{\infty}})$ and vertical exterior derivative $d_{\scaleto{\mkern-1.0muV}{4pt}}:\Omega^{r,s}({{\mathcal{R}}^{\infty}})\to\Omega^{r,s+1}({{\mathcal{R}}^{\infty}})$ are computed from the equations,

[TABLE]

The horizontal and vertical differentials satisfy

[TABLE]

The structure equations of ${\mathcal{I}}$ are computed using 2.11 to be

[TABLE]

Since $d_{\scaleto{\mkern-1.5muH}{4pt}}^{2}=0$, the complex $d_{\scaleto{\mkern-1.5muH}{4pt}}:\Omega^{r,s}({{\mathcal{R}}^{\infty}})\to\Omega^{r+1,s}({{\mathcal{R}}^{\infty}})$ is a differential complex and $H^{r,s}({{\mathcal{R}}^{\infty}})$ is its cohomology,

[TABLE]

The conservation laws of $\Delta$ are the $d_{\scaleto{\mkern-1.5muH}{4pt}}$ closed forms in $\Omega^{1,0}({{\mathcal{R}}^{\infty}})$ and $H^{1,0}({{\mathcal{R}}^{\infty}})$ is the space of equivalence classes of conservation laws modulo the horizontal derivative of a function $d_{\scaleto{\mkern-1.5muH}{4pt}}f$, $f\in C^{\infty}({{\mathcal{R}}^{\infty}})$.

The vertical complex $d_{\scaleto{\mkern-1.0muV}{4pt}}:\Omega^{r,s}({{\mathcal{R}}^{\infty}})\to\Omega^{r,s+1}({{\mathcal{R}}^{\infty}})$ is a differential complex whose cohomology is trivial [anderson:2016a], [anderson-kamran:1997a]. Specifically, $d_{\scaleto{\mkern-1.0muV}{4pt}}$ is the ordinary exterior derivative in the variables $u_{i}$, and the DeRham homotopy formula (in $u_{i}$ variables with parameter) applies. The property $d_{\scaleto{\mkern-1.5muH}{4pt}}d_{\scaleto{\mkern-1.0muV}{4pt}}=-d_{\scaleto{\mkern-1.0muV}{4pt}}d_{\scaleto{\mkern-1.5muH}{4pt}}$ make $d_{\scaleto{\mkern-1.0muV}{4pt}}:H^{r,s}({{\mathcal{R}}^{\infty}})\to H^{r,s+1}({{\mathcal{R}}^{\infty}})$ a co-chain map up to sign, see Appendix A.

Remark 2.1**.**

Every function $Q(t,x,u,u_{x},u_{xx},\ldots,u_{k})$ on $J^{\infty}(\mathbb{R}^{2},\mathbb{R})$ factors through $\pi:J^{\infty}(\mathbb{R}^{2},\mathbb{R})\to{{\mathcal{R}}^{\infty}}$, $\pi(t,x,u,u_{t},u_{x},u_{tt},u_{tx},u_{xx},\ldots)=(t,x,u,u_{x},u_{xx},\ldots)$ which is a left inverse of $\iota$ in equation 2.9. Therefore, by an abuse of notation, we view a function $Q(t,x,u,u_{x},u_{xx},\ldots,u_{k})$ either on $J^{\infty}(\mathbb{R}^{2},\mathbb{R})$ or ${{\mathcal{R}}^{\infty}}$ where the context will determine which. For example,

[TABLE]

is a differential operator on $J^{\infty}(\mathbb{R}^{2},\mathbb{R})$ while

[TABLE]

is a differential operator on ${{\mathcal{R}}^{\infty}}$ where $\pi_{*}({\bf F}_{Q})={\bf L}_{Q}$ . A differential operator $\bar{\mathcal{E}}=r_{i}(t,x,u,u_{x},\ldots)X^{i}$ on ${{\mathcal{R}}^{\infty}}$ lifts to ${\mathcal{E}}=r_{i}D_{x}^{i}$ and $\pi_{*}{\mathcal{E}}=\bar{\mathcal{E}}$. The formal adjoint of $\bar{\mathcal{E}}$ acting on a form $\omega$ is $(-X_{i})^{i}(r_{i}\omega)$. The operator $\bar{\mathcal{E}}$ is skew-adjoint if and only if ${\mathcal{E}}$ is skew adjoint.

3. Canonical forms for $H^{1,s}({{\mathcal{R}}^{\infty}})$ and characteristic forms

The universal linearization (see [anderson-kamran:1997a]) of $\Delta=u_{t}-K(t,x,u,u_{x},\ldots,u_{n})$ on ${{\mathcal{R}}^{\infty}}$ is the differential operator (on ${{\mathcal{R}}^{\infty}}$),

[TABLE]

where $K_{i}=\partial_{u_{i}}K$, and the vector fields $T$ and $X$ are defined in equation 2.10. The operator ${\bf L}_{\Delta}$ is the restriction of the Fréchet derivative of $\Delta$ to ${{\mathcal{R}}^{\infty}}$. The adjoint of ${\bf L}_{\Delta}$ is given by

[TABLE]

This next theorem provides a normal form for a representative of the cohomology classes in $H^{1,s}({{\mathcal{R}}^{\infty}})$ and is analogous to Theorem 5.1 in [anderson-kamran:1997a].

Theorem 3.1**.**

Let $u_{t}=K(t,x,u,u_{x},\ldots,u_{n})$ be an $n^{th}$ order evolution equation and $H^{r,s}({{\mathcal{R}}^{\infty}})$ its cohomology. For any $[\omega]\in H^{1,s}({{\mathcal{R}}^{\infty}})$ there exists a representative,

[TABLE]

where $\rho\in\Omega^{0,s-1}({\mathcal{R}}^{\infty}),\ \beta\in\Omega^{0,s}({\mathcal{R}}^{\infty})$ and ${{\bf L}}_{\Delta}^{*}(\rho)=0$.

Proof.

The proof follows Theorem 5.1 of [anderson-kamran:1997a]. Choose $\tilde{\omega}_{0}\in\Omega^{1,s}(J^{\infty}(\mathbb{R}^{2},\mathbb{R}))$ such that

[TABLE]

where $\iota:{{\mathcal{R}}^{\infty}}\to J^{\infty}(\mathbb{R}^{2},\mathbb{R})$ is given in equation 2.9. Since $\iota^{*}(d_{\scaleto{\mkern-1.5muH}{4pt}}\tilde{\omega}_{0})=0$, it there exists $\tilde{\zeta}_{ab}\in\Omega^{0,s}(J^{\infty}(\mathbb{R}^{2},\mathbb{R})),\tilde{\mu}_{ab}\in\Omega^{0,s-1}(J^{\infty}(\mathbb{R}^{2},\mathbb{R}))$ such that (see Lemma 5.2 in [anderson-kamran:1997a])

[TABLE]

Applying the identical integration by parts argument on page 292 [anderson-kamran:1997a] to 3.4, implies there exists $\tilde{\zeta}\in\Omega^{0,s}(J^{\infty}(\mathbb{R}^{2},\mathbb{R})),\tilde{\rho}\in\Omega^{0,s-1}(J^{\infty}(\mathbb{R}^{2},\mathbb{R}))$ and $\tilde{\omega}\in\Omega^{1,s}(J^{\infty}(\mathbb{R}^{2},\mathbb{R}))$ such that $\tilde{\omega}=\tilde{\omega}_{0}+\tilde{d}_{\scaleto{\mkern-1.5muH}{4pt}}\tilde{\eta}$ and $\iota^{*}\tilde{\eta}=0$ (hence $\iota^{*}\tilde{\omega}=\omega$) and where

[TABLE]

We now apply $\iota^{*}\circ J$ to equation 3.5, where $J$ is defined in equation 2.4. For the first term in right hand side of equation 3.5 we find

[TABLE]

since each term contains a total derivative of $\Delta$, and these vanish under pullback to ${{\mathcal{R}}^{\infty}}$.

We now apply $\iota^{*}\circ J$ to the second term in the right hand side of 3.5,

[TABLE]

because $\iota^{*}(D_{t}^{a}D_{x}^{i}d_{\scaleto{\mkern-1.0muV}{4pt}}\Delta)=0$. Now

[TABLE]

with all other $\partial_{u_{a,i}}\mathbin{\raise 0.5pt\hbox{\hbox{{\vbox{\hrule height=0.4pt,width=6.0pt,depth=0.0pt}}}\vrule height=6.0pt,width=0.4pt,depth=0.0pt}\,}d_{\scaleto{\mkern-1.0muV}{4pt}}\Delta=0$, so that equation 3.7 becomes,

[TABLE]

By equation 2.5 $J(d_{\scaleto{\mkern-1.5muH}{4pt}}\tilde{\omega})=0$, so that applying $\iota^{*}\circ J$ to equation 3.5 implies $\iota^{*}J(d_{\scaleto{\mkern-1.0muV}{4pt}}\Delta\wedge\tilde{\rho})=0$, and so equation 3.8 gives ${\bf L}_{\Delta}^{*}(\iota^{*}\tilde{\rho})=0$.

We now turn to showing that equation 3.2 holds using the horizontal homotopy operator (equations 5.15, 5.16 and below 5.16 in [anderson-kamran:1997a]), see also proposition 4.12 page 117 of [anderson:2016a] or equation 5.133 in [Olver:1993a]. Using the notation $h^{r,s}_{\scaleto{\mkern-1.5muH}{4pt}}$ from [anderson:2016a], this operator satisfies

[TABLE]

Applying the pull back by $\iota$ to this formula gives the representative for $[\omega]$,

[TABLE]

with $d_{\scaleto{\mkern-1.5muH}{4pt}}\tilde{\omega}$ in 3.5.

To utilize the formula in [anderson:2016a] for $h^{2,s}_{\scaleto{\mkern-1.5muH}{4pt}}$ let $(x^{1}=t,x^{2}=x)$ and so for example $\vartheta^{1122}=D_{x}D_{x}D_{t}D_{t}\vartheta^{0}$ and let $k$ be the max of $|I|$ (number of derivatives) of $\vartheta^{I}$ terms in $(\vartheta_{t}-K_{i}\vartheta^{i})\wedge\rho$. Then by definition 4.13 on page 117 in [anderson:2016a] (or 5.134 in [Olver:1993a])

[TABLE]

where $(d_{\scaleto{\mkern-1.5muH}{4pt}}\tilde{\omega})_{j}=D_{x^{j}}\mathbin{\raise 0.5pt\hbox{\hbox{{\vbox{\hrule height=0.4pt,width=6.0pt,depth=0.0pt}}}\vrule height=6.0pt,width=0.4pt,depth=0.0pt}\,}d_{\scaleto{\mkern-1.5muH}{4pt}}\tilde{\omega}=(-1)^{j+1}d\hat{x}_{j}\wedge(\Delta\tilde{\zeta}+d_{\scaleto{\mkern-1.0muV}{4pt}}\Delta\wedge\tilde{\rho})$, $(\hat{x}_{1}=x,\hat{x}_{2}=t)$, $I=(i_{1},\ldots,i_{l})$, $|I|=l$, and

[TABLE]

Applying $\iota^{*}$ to equation 3.10 we have the $\iota^{*}h^{2,s}_{\scaleto{\mkern-1.5muH}{4pt}}\left(dt\wedge dx\wedge(\Delta\tilde{\zeta})\right)=0$ because all terms in 3.11 on $\Delta\tilde{\zeta}$ involve total derivatives of $\Delta$. Therefore using equation 3.10, equation 3.9 becomes,

[TABLE]

Consider the first term in equation 3.12. The only non-zero interior product is (with $u_{(1)}=u_{t},u_{(2)}=u_{x}$ etc.)

[TABLE]

since $\Delta$ does not depend on derivatives such as $u_{tx}$. Therefore the only non-zero terms have $|I|=0,|L|=0$ in the first term of 3.12 giving,

[TABLE]

Combining equation 3.13 with 3.12 we have

[TABLE]

which produces equation 3.2 with $\rho=\frac{1}{s}\iota^{*}\tilde{\rho}$ . Equations 3.14 and 3.8 shows that $\rho=\frac{1}{s}\iota^{*}\tilde{\rho}$ satisfies ${\bf L}_{\Delta}^{*}(\rho)=0$. ∎

If $s=1$ in Theorem 3.1, then $\rho\in C^{\infty}({{\mathcal{R}}^{\infty}})$ and is the characteristic function for the cohomology class $[\omega]\in H^{1,1}({{\mathcal{R}}^{\infty}})$, see Theorem 3.4 and Theorem A.1. In general $\rho$ in equation 3.2 is called a characteristic form for $[\omega]$ see [anderson-kamran:1997a]. The form $\beta$ in 3.2 is given in terms of $\rho$ by formula 3.14 which is simplified in Theorem 3.4 for $H^{1,1}({{\mathcal{R}}^{\infty}})$ and $H^{1,2}({{\mathcal{R}}^{\infty}})$. The term $dx\wedge\theta^{0}\wedge\rho$ in equation 3.2 generalizes the conserved density of a conservation law, and plays a critical role in Section 7.

Theorem 3.2**.**

For $s\geq 3$, $H^{1,s}({{\mathcal{R}}^{\infty}})=0$.

Proof.

Suppose $\omega$ is a representative for an element of $H^{1,s+1}({{\mathcal{R}}^{\infty}})$, $(s\geq 2)$, in the form 3.2, where $\rho\in\Omega^{0,s}({{\mathcal{R}}^{\infty}})$ is given by

[TABLE]

and satisfies ${\bf L}_{\Delta}^{*}(\rho)=0$.

Suppose for $\rho$ in 3.15 that the highest form order for (no sum) $A_{m_{1}\ldots m_{s}}\theta^{m_{1}}\wedge\theta^{m_{2}}\wedge\ldots\theta^{m_{s}}$ where we use max of ($m_{1}>m_{2}>\ldots>m_{s}$) to determine highest order. We first claim that in ${\bf L}_{\Delta}^{*}(\rho)$ the coefficient of $\theta^{m_{1}+n}\wedge\theta^{m_{2}}\wedge\ldots\theta^{m_{s}}$ is

[TABLE]

and that the coefficient of $\theta^{m_{1}+n-1}\wedge\theta^{m_{1}}\wedge\ldots\theta^{m_{s}}$ when $m_{1}>m_{2}+1$ is

[TABLE]

and when $m_{1}=m_{2}+1$ and $n$ is odd, the coefficient of $\theta^{m_{1}+n-1}\wedge\theta^{m_{1}}\wedge\ldots\theta^{m_{s}}$

[TABLE]

Therefore if ${\bf L}_{\Delta}^{*}(\rho)=0$ implies $A_{m_{1}\ldots m_{s}}=0$, then $\rho=0$ and $\omega=dt\wedge\beta$. The condition $d_{\scaleto{\mkern-1.5muH}{4pt}}\omega=0$ gives $X(\beta)=0$. This implies $\beta=0$ since $\beta\in\Omega^{0,s+1}({{\mathcal{R}}^{\infty}})$ and so $\omega=0$.

We compute ${\bf L}_{\Delta}^{*}(\rho)=-T(\rho)-(-X)^{i}(K_{i}\rho)$ to find equations 3.16,3.17,3.18,

[TABLE]

where from equation 2.15 we have $T(\theta^{i})=K_{n}\theta^{i+n}+{\rm lower\ order}$. Consider also the highest order terms in while expanding $X^{n}(K_{n}A_{i_{1}\ldots i_{s}}\theta^{i_{1}}\wedge\ldots\theta^{i_{s}})$,

[TABLE]

The coefficient of $\theta^{m_{1}+n}\wedge\theta^{m_{2}}\wedge\ldots\theta^{m_{s}}$ (which is the highest order) occurring in equation 3.19 comes from the second term on the right hand side in 3.19 and the first term on the last right hand side in equation 3.20 to give 3.16.

We consider the next highest order term in 3.19. From equation 3.19, the only possible term that can contain $\theta^{m_{1}+n-1}\wedge\theta^{m_{2}+1}\wedge\theta^{m_{3}}\ldots\theta^{m_{s}}$ when $m_{1}>m_{2}+1$ is from second term on the last right hand side of equation 3.20. Therefore 3.20 produces 3.17.

In the case when $m_{1}=m_{2}+1$ we have the third term in 3.19 at highest order giving

[TABLE]

then using $m_{1}=m_{2}+1$ the first term equals

[TABLE]

From the other two terms on the last right hand side in equation 3.20 we have the two terms

[TABLE]

Combining equations 3.22 and using $m_{1}=m_{2}+1$ and $n$ is odd in equation 3.23, gives equation 3.18. ∎

We also have as a corollary of Theorem 3.1.

Corollary 3.3**.**

If $u_{t}=K(t,x,u,\ldots,u_{2m})$ is an even order evolution equation, then $H^{1,2}({{\mathcal{R}}^{\infty}})=0$.

Proof.

We show that the only $\rho\in\Omega^{0,1}({{\mathcal{R}}^{\infty}})$ satisfying ${\bf L}_{\Delta}^{*}(\rho)=0$ is $\rho=0$. Suppose $\rho=r_{i}\theta^{i},\ i=0,\ldots,k$ then by direct computation,

[TABLE]

which is non-zero unless $r_{k}=0$. Therefore $\rho=0$. ∎

We now refine Theorem 3.1 which gives a formula for $\beta$ in equation 3.2.

Theorem 3.4**.**

Let $u_{t}=K(t,x,u,u_{x},\ldots,u_{n})$ be an $n^{th}$ order evolution equation. For any $[\omega]\in H^{1,s}({{\mathcal{R}}^{\infty}})$, $s=1,2$ there exists a representative, $\omega\in\Omega^{1,s}({\mathcal{R}}^{\infty})$, $s=1,2$ such that

[TABLE]

where $\rho\in\Omega^{0,s-1}({\mathcal{R}}^{\infty})$ ($s=1,2$) satisfies $\theta^{0}\wedge{\bf L}_{\Delta}^{*}(\rho)=0$. If $s=1$ then $\rho\in C^{\infty}({{\mathcal{R}}^{\infty}})$ and the representative 3.24 is unique.

Proof.

First suppose $\rho\in\Omega^{0,s-1}({{\mathcal{R}}^{\infty}})$ and satisfies $\theta^{0}\wedge{\bf L}_{\Delta}^{*}(\rho)=0$, and let $\omega\in\Omega^{1,s}({{\mathcal{R}}^{\infty}})$ be as in equation 3.24. We compute $d_{\scaleto{\mkern-1.5muH}{4pt}}\omega$,

[TABLE]

To compute $X({\bm{\beta}}(\rho))$ we need the telescoping identity,

[TABLE]

Using equation 3.26 in the formula for ${\bm{\beta}}(\rho)$ in equation 3.24 gives

[TABLE]

We then use

[TABLE]

so that together with equation 3.27, equation 3.25 becomes (adding and subtracting $K_{0}\theta^{0}\wedge\rho$)

[TABLE]

Now by equation 2.7, $\iota^{*}d_{\scaleto{\mkern-1.0muV}{4pt}}(-\Delta)=-T(\theta^{0})+\sum_{i=0}^{n}K_{i}\theta^{i}=0$, and so equation 3.28 becomes,

[TABLE]

Now by Theorem 3.4 there exists representative $\hat{\omega}=dx\wedge\theta^{0}\wedge\rho-dt\wedge\beta$ where ${\bf L}^{*}(\rho)=0$. Let $\omega$ be the form in 3.24 using this $\rho$. The form $\omega^{\prime}=\hat{\omega}-\omega$ satisfies

[TABLE]

This implies $X(\beta-{\bm{\beta}}(\rho))=0$, where $\beta-{\bm{\beta}}(\rho)\in\Omega^{0,s}({{\mathcal{R}}^{\infty}}),s=1,2$. However, the only contact form satisfying this condition is the zero form. So $\beta={\bm{\beta}}(\rho)$. This proves equation 3.24.

For the final statement in the theorem, suppose $\omega_{a}=dx\wedge\theta^{0}\cdot Q_{a}-dt\wedge\beta_{a}$, $a=1,2$ where $Q_{a}\in C^{\infty}({{\mathcal{R}}^{\infty}})$ and $\beta_{a}\in\Omega^{0,1}({{\mathcal{R}}^{\infty}})$ satisfy $[\omega_{1}]=[\omega_{2}]\in H^{1,1}({{\mathcal{R}}^{\infty}})$. This implies there exists $\xi=g_{j}\theta^{j}$ such that $\omega_{1}-\omega_{2}=d_{\scaleto{\mkern-1.5muH}{4pt}}\xi$ so that

[TABLE]

Since $X(\theta^{i})=\theta^{i+1}$, equation 3.30 can only be satisfied when $Q_{1}=Q_{2}$ and $g_{j}=0$. Then the condition $d_{\scaleto{\mkern-1.5muH}{4pt}}(\omega_{1}-\omega_{2})=dx\wedge dt\wedge X(\beta_{1}-\beta_{2})=0$ implies $X(\beta_{1}-\beta_{2})=0$ which in turn implies $\beta_{1}=\beta_{2}$ because $\beta_{1}-\beta_{2}\in\Omega^{0,1}({{\mathcal{R}}^{\infty}})$. Therefore the form in equation 3.24 for $s=1$ is unique. ∎

The form $\omega$ in 3.24 was originally derived by a rather lengthy calculation of the second term in 3.14.

Corollary 3.5**.**

If $[\omega]\in H^{1,2}({{\mathcal{R}}^{\infty}})$ with $\epsilon=r_{i}\theta^{i},\ i=0,\dots,k$ and representative

[TABLE]

where $\epsilon^{*}=(-X)^{i}(r_{i}\theta^{0})=-\epsilon$, then ${\bf L}^{*}_{\Delta}(\epsilon)=0$ and $\beta={\bm{\beta}}(\epsilon)$ is given by equation 3.24.

Proof.

Write $\beta=B_{ab}\theta^{a}\wedge\theta^{b}$, and choose in equation 3.3, $\tilde{\omega}_{0}=dx\wedge\vartheta^{0}\wedge\tilde{\epsilon}_{0}-dt\wedge\tilde{\beta}$, where $\tilde{\epsilon}_{0}=r_{i}\vartheta^{i},\tilde{\beta}=B_{ab}\vartheta^{a}\wedge\vartheta^{b}$ (see Remark 2.1). Then there exists $s_{ab}\in C^{\infty}({J^{\infty}(\mathbb{R}^{2},\mathbb{R})})$ such that,

[TABLE]

since $D_{t}(\vartheta^{0})=d_{\scaleto{\mkern-1.0muV}{4pt}}\Delta+K_{m}\vartheta^{m}$. Now using equations 3.26 and 3.27 with $X=D_{x}$, $\rho=\vartheta^{0}$, $K_{i}=r_{i}$, and $\theta^{0}=d_{\scaleto{\mkern-1.0muV}{4pt}}\alpha$ while adding and subtracting $r_{0}\vartheta^{0}\wedge d_{\scaleto{\mkern-1.0muV}{4pt}}\Delta$ we have

[TABLE]

where

[TABLE]

and $\tilde{\eta}$ satisfies $\iota^{*}\tilde{\eta}_{0}=0$. Since $\epsilon^{*}=-\epsilon$, we have $\tilde{\epsilon}_{0}^{*}=-\tilde{\epsilon}_{0}$, and combining this with equation 3.31 and 3.32 we have

[TABLE]

Therefore comparing equations 3.34 with equation 3.5 we have $\tilde{\rho}=2\tilde{\epsilon}_{0}$. By equations 3.14 in the proof of Theorem 3.1, $\rho=\frac{1}{2}\iota^{*}\tilde{\rho}=\epsilon$ satisfies ${\bf L}^{*}_{\Delta}(\epsilon)=0$. Finally Theorem 3.4 implies $\beta={\bm{\beta}}(\epsilon)$. ∎

4. Canonical forms for $H^{1,2}({{\mathcal{R}}^{\infty}})$

By refining Theorem 3.1 we will produce a canonical form for elements of $H^{1,2}({{\mathcal{R}}^{\infty}})$, by determining a unique representative. A form $\rho\in\Omega^{0,1}({{\mathcal{R}}^{\infty}})$ can always be written $\rho=r_{i}X^{i}(\theta^{0})$. We define the adjoint of $\rho$ by $\rho^{*}=(-X)^{i}(r_{i}\theta^{0})$ while $(\rho^{*})^{*}=\rho$ because the operator $r_{i}X^{i}$ has this property, see Remark 2.1.

Theorem 4.1**.**

Let $[\omega]\in H^{1,2}({{\mathcal{R}}^{\infty}})$. There exists a unique representative having the form

[TABLE]

where $\epsilon^{*}=-\epsilon$, and ${\bm{\beta}}(\epsilon)$ is given by the formula in 3.24 and $\theta^{0}\wedge{\bf L}_{\Delta}^{*}(\epsilon)=0$.

Proof.

We begin by utilizing equation 3.27 and make the substitution $\rho=\theta^{0}$, $K_{i}=r_{i}$ giving the identity,

[TABLE]

If we now write $\rho=\sum_{i=1}^{k}r_{i}\theta^{i}$ and let

[TABLE]

the identity 4.2 gives

[TABLE]

Suppose now $[\omega]\in H^{1,2}({{\mathcal{R}}^{\infty}})$ with representative $\omega=dx\wedge\theta^{0}\wedge\rho-dt\wedge{\bm{\beta}}(\rho)$ with $\rho=r_{i}\theta^{i}$ from Theorem 3.2. Let $\hat{\omega}=\omega-\frac{1}{2}d_{\scaleto{\mkern-1.5muH}{4pt}}(\eta)$ so that $[\hat{\omega}]=[\omega]$ and where $\eta$ is given in equation 4.3. We then use equation 4.4 to replace $X(\eta)$ in the following,

[TABLE]

The representative $\hat{\omega}$ in 4.5 satisfies the skew adjoint condition in the theorem with

[TABLE]

while Corollary 3.5 shows ${\bm{\beta}}(\epsilon)={\bm{\beta}}(\rho)+\frac{1}{2}T(\eta)$.

We now show the representative 4.1 unique. Suppose that

[TABLE]

where $\epsilon_{\alpha}^{*}=-\epsilon_{\alpha}$ and $[\omega_{1}]=[\omega_{2}]$. This implies there exists $\xi=\xi_{ab}\theta^{a}\wedge\theta^{b}\in\Omega^{0,2}({{\mathcal{R}}^{\infty}})$ such that

[TABLE]

Now let $\tilde{\epsilon}_{\alpha}=r_{i,\alpha}\vartheta^{i}\in\Omega^{0,1}({J^{\infty}(\mathbb{R}^{2},\mathbb{R})})$ and $\tilde{\xi}=\xi_{ab}\vartheta^{a}\wedge\vartheta^{b}\in\Omega^{0,2}({J^{\infty}(\mathbb{R}^{2},\mathbb{R})})$ so that $\iota^{*}\tilde{\epsilon}_{\alpha}=\epsilon_{\alpha}$, $\iota^{*}\tilde{\xi}=\xi$. Equation 4.7 implies

[TABLE]

Applying the integration by parts operator $I$ (using 6.2) to equation 4.8 and that $\tilde{\epsilon}^{*}_{\alpha}=-\tilde{\epsilon}_{\alpha}$ gives

[TABLE]

Since $\tilde{\epsilon}_{1}-\tilde{\epsilon}_{2}$ is skew-adjoint, this implies $\tilde{\epsilon}_{1}=\tilde{\epsilon}_{2}$ and that $\epsilon_{1}=\epsilon_{2}$. This implies ${\bm{\beta}}(\epsilon_{1})={\bm{\beta}}(\epsilon_{2})$ and so $\omega_{1}=\omega_{2}$. ∎

Corollary 4.2**.**

If $[\omega]\in H^{1,2}({{\mathcal{R}}^{\infty}})$ with representative $\omega=dx\wedge\theta^{0}\wedge\rho-dt\wedge{\bm{\beta}}(\rho)$ then the unique representative in Theorem 4.1 has

[TABLE]

We now refine Theorem 3.1 and provide a third (non-unique) normal form.

Theorem 4.3**.**

Let $[\omega]\in H^{1,2}({{\mathcal{R}}^{\infty}})$ then there exists a representative $\omega$ such that

[TABLE]

where ${Q\,}$ is a smooth function on ${{\mathcal{R}}^{\infty}}$ and $\gamma\in\Omega^{0,1}({{\mathcal{R}}^{\infty}})$.

Proof.

We start with equation 3.2 in Theorem 3.1 where a representative for $[\omega]$ can be written

[TABLE]

where w.l.o.g. $\rho=r_{a}\theta^{a},a=1,\ldots,m$. Now $d_{\scaleto{\mkern-1.0muV}{4pt}}\omega\in H^{1,3}({{\mathcal{R}}^{\infty}})$, and so by Theorem 3.2 there exists $\xi\in\Omega^{0,3}({{\mathcal{R}}^{\infty}})$ such that

[TABLE]

Writing $\xi=A_{ijk}\theta^{i}\wedge\theta^{j}\wedge\theta^{k}$, this gives

[TABLE]

We now show $\xi$ has the form

[TABLE]

Suppose there is a term in $\xi$ with $\theta^{M_{1}}\wedge\theta^{M_{2}}\wedge\theta^{M_{3}}$, $1\leq M_{1}<M_{2}<M_{3}$, and assume we have the one with the highest $M_{3}$. On the left side of 4.11 there will be

[TABLE]

which contains $dx\wedge\theta^{M_{1}}\wedge\theta^{M_{2}}\wedge\theta^{M_{3}+1}$, which can’t occur on the right side since there is no $\theta^{0}$. Suppose now that there are terms in $\xi$ of the form $\theta^{0}\wedge\theta^{M_{2}}\wedge\theta^{M_{3}}$ with $1<M_{2}<M_{3}$. Consider the maximal $M_{3}$, and again

[TABLE]

will contain a term $dx\wedge\theta^{0}\wedge\theta^{M_{2}}\wedge\theta^{M_{3}+1}$ which can’t occur on the right hand side of equation 4.11. This shows equation 4.12.

Now

[TABLE]

but since $\xi\in\Omega^{0,3}({{\mathcal{R}}^{\infty}})$, this implies $d_{\scaleto{\mkern-1.0muV}{4pt}}\xi=0$. We apply vertical exactness, and let $\zeta\in\Omega^{0,2}({{\mathcal{R}}^{\infty}})$ be such that $d_{\scaleto{\mkern-1.0muV}{4pt}}\zeta=\xi$. By the vertical homotopy on $\xi$ we may assume $\zeta=A\theta^{0}\wedge\theta^{1}$ Finally, we let

[TABLE]

where $\tilde{\rho}=\rho+X(A\theta^{1}\wedge)$ and $\tilde{\beta}=\beta-T(\zeta)$. Therefore,

[TABLE]

This proves there is a representative $\tilde{\omega}$ for $[\omega]$ with $d_{\scaleto{\mkern-1.0muV}{4pt}}\tilde{\omega}=0$.

Again we use $d_{\scaleto{\mkern-1.0muV}{4pt}}$ exactness to find $\eta\in\Omega^{1,1}({{\mathcal{R}}^{\infty}})$ such that,

[TABLE]

where

[TABLE]

Writing $\alpha=a_{j}\theta^{j}$, $j=0,\ldots,m$, equation 4.14 and 4.15 give

[TABLE]

We now modify $\eta$ in equation 4.15 and the representative $\tilde{\omega}$ for $[\omega]$ in equation 4.13 by

[TABLE]

so that $\hat{\omega}=d_{\scaleto{\mkern-1.0muV}{4pt}}\hat{\eta}$. In particular we note

[TABLE]

Continuing by induction, there exists a representative $\bar{\omega}$ for $[\omega]$ and an $\bar{\eta}\in\Omega^{1,1}({{\mathcal{R}}^{\infty}})$, where $\bar{\omega}=d_{\scaleto{\mkern-1.0muV}{4pt}}\bar{\eta}$ and

[TABLE]

where ${Q\,}$ is a smooth function on ${{\mathcal{R}}^{\infty}}$. This also implies

[TABLE]

∎

Combining Theorem 4.3 and Corollary 4.2 gives the following.

Corollary 4.4**.**

If $[\omega]\in H^{1,2}({{\mathcal{R}}^{\infty}})$ with representative $\omega=d_{\scaleto{\mkern-1.0muV}{4pt}}(dx\wedge\theta^{0}\cdot Q-dt\wedge\gamma)$ from Theorem 4.3 then the unique representative in Theorem 4.1 is determined by

[TABLE]

where ${\bf L}_{Q}=Q_{i}X^{i}$.

The snake lemma from the variational bicomplex is the following.

Lemma 4.5**.**

Let $\omega\in\Omega^{1,2}({{\mathcal{R}}^{\infty}})$ and $d_{\scaleto{\mkern-1.5muH}{4pt}}\omega=0$, $d_{\scaleto{\mkern-1.0muV}{4pt}}\omega=0$. Let $\eta\in\Omega^{1,1}({{\mathcal{R}}^{\infty}})$ such that $d_{\scaleto{\mkern-1.0muV}{4pt}}\eta=\omega$. Then there exists $\lambda=L\,dt\wedge dx\in\Omega^{2,0}({{\mathcal{R}}^{\infty}})$ such that $d_{\scaleto{\mkern-1.5muH}{4pt}}\eta=d_{\scaleto{\mkern-1.0muV}{4pt}}\lambda$.

Proof.

We have

[TABLE]

and the vertical exactness of the bicomplex immediately implies the lemma. ∎

The relationship between $\omega$, $\lambda$ and $\eta$ in Lemma 4.5 is represented by the diagram,

[TABLE]

Corollary 4.6**.**

Let $[\omega]\in H^{1,2}({{\mathcal{R}}^{\infty}})$ and let $\omega$ be the $d_{\scaleto{\mkern-1.0muV}{4pt}}$ closed representative from equation 4.10, and let $\lambda\in\Omega^{2,0}({{\mathcal{R}}^{\infty}})$ be as in Lemma 4.5, so that $d_{\scaleto{\mkern-1.5muH}{4pt}}\eta=d_{\scaleto{\mkern-1.0muV}{4pt}}\lambda$. The linear map $\Lambda:H^{1,2}({{\mathcal{R}}^{\infty}})\to H^{2,0}({{\mathcal{R}}^{\infty}})$ given by

[TABLE]

is well defined.

Proof.

Suppose $\omega_{a}\in\Omega^{1,2}({{\mathcal{R}}^{\infty}}),a=1,2$ where $[\omega_{1}]=[\omega_{2}]$ and that $\omega_{a}=d_{\scaleto{\mkern-1.0muV}{4pt}}\eta_{a},a=1,2$ are two $d_{\scaleto{\mkern-1.0muV}{4pt}}$ closed representatives. Let $\lambda_{a}\in\Omega^{2,0}({{\mathcal{R}}^{\infty}})$ satisfy $d_{\scaleto{\mkern-1.5muH}{4pt}}\eta_{a}=d_{\scaleto{\mkern-1.0muV}{4pt}}\lambda_{a},a=1,2$. Since $[\omega_{1}]=[\omega_{2}]$, there exists $\xi\in\Omega^{0,2}({{\mathcal{R}}^{\infty}})$ such that,

[TABLE]

This implies $d_{\scaleto{\mkern-1.0muV}{4pt}}d_{\scaleto{\mkern-1.5muH}{4pt}}\xi=-d_{\scaleto{\mkern-1.5muH}{4pt}}d_{\scaleto{\mkern-1.0muV}{4pt}}\xi=0$ which then implies $d_{\scaleto{\mkern-1.0muV}{4pt}}\xi=0$ since $H^{0,2}({{\mathcal{R}}^{\infty}})=0$ (or $d_{\scaleto{\mkern-1.5muH}{4pt}}\mu=0,\mu\in\Omega^{0,s}({{\mathcal{R}}^{\infty}})$ implies $\mu=0$). Therefore there exists $\phi\in\Omega^{0,1}({{\mathcal{R}}^{\infty}})$ such that

[TABLE]

which implies

[TABLE]

and that

[TABLE]

By vertical exactness there exists $\kappa\in\Omega^{1,1}({{\mathcal{R}}^{\infty}})$ such that

[TABLE]

Taking $d_{\scaleto{\mkern-1.5muH}{4pt}}$ of this equation gives

[TABLE]

so

[TABLE]

Therefore

[TABLE]

where $\mu\in\Omega^{2}(\mathbb{R}^{2})$. The deRham cohomology of $\mathbb{R}^{2}$ is trivial so $\mu=d\alpha=d_{\scaleto{\mkern-1.5muH}{4pt}}\alpha,\alpha\in\Omega^{1}(\mathbb{R}^{2})$. Therefore equation 4.19 becomes

[TABLE]

and $[\lambda_{1}]=[\lambda_{2}]\in H^{2,0}({{\mathcal{R}}^{\infty}})$. ∎

The relevance of the kernel of $\Lambda$ is given in Theorem A.4.

5. Variational Operators and $H^{1,2}({{\mathcal{R}}^{\infty}})$

A scalar evolution equation with $\Delta=u_{t}-K(t,x,u,u_{x},\ldots,u_{n})$ is said to admit a variational operator of order $k$ if there exists a differential operator

[TABLE]

$r_{k}\neq 0$, and a function $L\in C^{\infty}(\mathbb{R}^{2},\mathbb{R})$ such that,

[TABLE]

where ${\bf E}(L)$ is the Euler-Lagrange expression of $L\in C^{\infty}(J^{\infty}(\mathbb{R}^{2},\mathbb{R}))$. We will prove a generalization of Theorem 2.6 in [anderson-thompson:1992a] which relates the existence of a multiplier (or zero order operator) to the cohomology of $\Delta$. We start with the following theorem.

Theorem 5.1**.**

Let ${{\mathcal{E}}}=r_{i}D_{x}^{i}$ be a variational operator for $\Delta=u_{t}-K(t,x,u,u_{x},\ldots,u_{n})$ with Lagrangian $L$ satisfying 5.1. Then there exists $\eta\in\Omega^{1,1}(J^{\infty}(\mathbb{R}^{2},\mathbb{R}))$ such that

[TABLE]

With $\iota:{{\mathcal{R}}^{\infty}}\to J^{\infty}(\mathbb{R}^{2},\mathbb{R})$ in equation 2.9 let,

[TABLE]

Then $d_{\scaleto{\mkern-1.5muH}{4pt}}\omega=0$.

Proof.

Suppose ${\mathcal{E}}$ and $L$ are given satisfying 5.1, then using the standard formula in the calculus of variations (for example equation (3.2) in [anderson:1992a]), we have on account of 5.1

[TABLE]

which shows 5.2.

By applying $\iota^{*}$ to equation 5.2 we have

[TABLE]

Letting $\omega=d_{\scaleto{\mkern-1.0muV}{4pt}}\iota^{*}\eta$, we compute $d_{\scaleto{\mkern-1.5muH}{4pt}}\omega$ using equation 5.4 and get

[TABLE]

Therefore $d_{\scaleto{\mkern-1.5muH}{4pt}}\omega=0$. ∎

A formula for $\omega$ in terms of ${\mathcal{E}}$ in Theorem 5.1 is given in Theorem 5.3 just below. Before giving the theorem we note the following property of variational operators for evolutions equations.

Lemma 5.2**.**

If $\Delta=u_{t}-K(t,x,u,u_{x},\ldots,u_{n})$ admits the $k^{th}$ order variational operator ${{\mathcal{E}}}=r_{i}(t,x,u,u_{x},\ldots)D_{x}^{i},\ i=0,\ldots,k$ then ${{\mathcal{E}}}$ is skew adjoint.

Proof.

Suppose ${{\mathcal{E}}}(u_{t}-K)={\bf E}(L)$ then applying $I\circ d_{\scaleto{\mkern-1.0muV}{4pt}}$ to equation 5.2 using $d^{2}_{\scaleto{\mkern-1.0muV}{4pt}}=0$ along with the property 2.5 for $I$, we have

[TABLE]

With $\Delta=u_{t}-K$ let

[TABLE]

so that condition 5.5 gives

[TABLE]

In the term $\partial_{u_{a,j}}\mathbin{\raise 0.5pt\hbox{\hbox{{\vbox{\hrule height=0.4pt,width=6.0pt,depth=0.0pt}}}\vrule height=6.0pt,width=0.4pt,depth=0.0pt}\,}\kappa$ where $\kappa$ is given in equation 5.6 we note that $\partial_{u_{a,j}}(r_{j})=0,\partial_{u_{a,j}}(K)=0,a\geq 1,j\geq 0$. Therefore the only possible non-zero terms in the summation term in equation 5.7 with $\partial_{u_{a,j}}\mathbin{\raise 0.5pt\hbox{\hbox{{\vbox{\hrule height=0.4pt,width=6.0pt,depth=0.0pt}}}\vrule height=6.0pt,width=0.4pt,depth=0.0pt}\,}$ with $a\geq 1,j\geq 0$ satisfy

[TABLE]

Writing the condition $I(dt\wedge dx\wedge\vartheta^{0}\wedge\kappa)\mod\{\vartheta^{j}\}_{j\geq 0}$ using equation 5.7 and 5.8 gives

[TABLE]

In order for the right side of equation 5.9 to be zero we must have ${{\mathcal{E}}}^{*}=-{{\mathcal{E}}}$. ∎

Theorem 5.3**.**

Let ${\mathcal{E}}=r_{i}(t,x,u,u_{x},\ldots)D_{x}^{i},\ i=0,\ldots,k$ be a $k^{th}$ order variational operator for $\Delta=u_{t}-K(t,x,u,u_{x},\ldots,u_{n})$ and let $[d_{\scaleto{\mkern-1.0muV}{4pt}}\iota^{*}\eta]\in H^{1,2}({{\mathcal{R}}^{\infty}})$ from Theorem 5.1. Then the unique representative for $[d_{\scaleto{\mkern-1.0muV}{4pt}}\iota^{*}\eta]$ in Theorem 4.1 is

[TABLE]

Proof.

Let $\hat{\omega}_{0}=d_{\scaleto{\mkern-1.0muV}{4pt}}\tilde{\eta}$ where $\eta$ satisfies equation 5.3. We have from equations 5.3 and 3.32

[TABLE]

where $\tilde{\eta}$ is given in equation 3.33 and satisfies $\iota^{*}\tilde{\eta}=0$.

As remarked in the proof of Theorem 3.1, the term $D_{x}^{i}(\Delta)d_{\scaleto{\mkern-1.0muV}{4pt}}r_{i}\wedge\vartheta^{0}$ in 5.11 does not contribute to the form $\tilde{\rho}$ in equation 3.5. Therefore we have from equation 5.11 that $\tilde{\rho}$ in equation 3.5 is,

[TABLE]

Since $\rho=\frac{1}{2}\iota^{*}\tilde{\rho}$ and ${\mathcal{E}}$ is skew-adjoint we get equation 5.10. ∎

We now come to the last main theorem in this section which proves the converse to Theorem 5.1. The proof is again a generalization of the argument given in Theorem 2.6 of [anderson-thompson:1992a] for the multiplier problem.

Theorem 5.4**.**

Let $[\omega]\in H^{1,2}({{\mathcal{R}}^{\infty}})$ with representative $\omega$ as in equation 4.10 in Theorem 4.3 given by

[TABLE]

where

[TABLE]

Let $\lambda=Ldt\wedge dx$ satisfying $d_{\scaleto{\mkern-1.5muH}{4pt}}\eta=d_{\scaleto{\mkern-1.0muV}{4pt}}\lambda$ from Lemma 4.5. Then ${\mathcal{E}}={\bf F}_{Q}^{*}-{\bf F}_{Q}$ is a variational operator and,

[TABLE]

The proof requires considerable care whether we are working on ${{\mathcal{R}}^{\infty}}$ or $J^{\infty}(\mathbb{R}^{2},\mathbb{R})$, see Remark 2.1.

Proof.

We start by writing $\gamma=g_{j}\theta^{j}$ in equation 5.13 and define the form $\tilde{\eta}\in J^{\infty}(\mathbb{R}^{2},\mathbb{R}))$ given by

[TABLE]

where $\iota^{*}\tilde{\eta}=\eta$ in equation 5.13, and the forms $\vartheta^{j}$ are defined in 2.1. Now define the vector fields on $J^{\infty}(\mathbb{R}^{2},\mathbb{R})$,

[TABLE]

so that $D_{t}=\tilde{T}+V,D_{x}=\tilde{X}+W$. Then

[TABLE]

Since $g_{j}=g_{j}(t,x,u,u_{x},\ldots)$ then $W(g_{j})=0$, while $dt\wedge dx\wedge D_{x}(\vartheta^{j})=dt\wedge dx\wedge\tilde{X}(\vartheta^{j})$ and $V(\vartheta^{0})=d_{\scaleto{\mkern-1.0muV}{4pt}}\Delta$. Equation 5.16 then can be written,

[TABLE]

The condition $d_{\scaleto{\mkern-1.5muH}{4pt}}\eta=d_{\scaleto{\mkern-1.0muV}{4pt}}\lambda$ (on ${{\mathcal{R}}^{\infty}}$) is

[TABLE]

on ${{\mathcal{R}}^{\infty}}$. Now $\pi^{*}dt\wedge dx\wedge\theta^{j}=dt\wedge dx\wedge\vartheta^{j}$ (see Remark 2.1), and using the vector fields in 5.15 we have

[TABLE]

Therefore applying $\pi^{*}$ to 5.18 and using 5.19 we have

[TABLE]

The first variational formula for $d_{\scaleto{\mkern-1.0muV}{4pt}}(Ldt\wedge dx)$ on $J^{\infty}(\mathbb{R}^{2},\mathbb{R})$ applied to the right side of 5.20 gives

[TABLE]

where $\tilde{\zeta}\in\Omega^{1,1}(J^{\infty}(\mathbb{R}^{2},\mathbb{R}))$. Inserting equation 5.21 into 5.17 we have,

[TABLE]

The terms $d_{\scaleto{\mkern-1.0muV}{4pt}}\Delta\cdot Q$ in equation 5.22 can be written as

[TABLE]

We now apply the integration by parts operator (see equation 2.8 in [anderson:1992a]) and use the first variational formula for $d_{\scaleto{\mkern-1.0muV}{4pt}}(Q\Delta dt\wedge dx)$, in equation 5.23 and get

[TABLE]

Next we expand the term $V(Q)$ in equation 5.22 using $V$ in 5.15 and

[TABLE]

Inserting 5.24, and 5.25 into 5.22 and letting $\tilde{\zeta}=\tilde{\zeta}_{1}+\tilde{\zeta}_{2}$ gives,

[TABLE]

This implies $d_{\scaleto{\mkern-1.5muH}{4pt}}(\tilde{\eta}-\tilde{\zeta})$ is a source-form. This is only possible if $d_{\scaleto{\mkern-1.5muH}{4pt}}(\tilde{\eta}-\tilde{\zeta})=0$, and so

[TABLE]

which is equation 5.14 as required. ∎

Remark 5.5**.**

In general three applications of the vertical homotopy operator are required to determine $\lambda\in\Omega^{2,0}({{\mathcal{R}}^{\infty}})$ from $[\omega]\in H^{1,2}({{\mathcal{R}}^{\infty}})$. The first is to find a representative $\omega\in H^{1,2}({{\mathcal{R}}^{\infty}})$ with $d_{\scaleto{\mkern-1.0muV}{4pt}}\omega=0$ (Theorem 4.3). The second is to find $\eta$ such that $d_{\scaleto{\mkern-1.0muV}{4pt}}\eta=\omega$, and the third is to find $\lambda$ such that $d_{\scaleto{\mkern-1.0muV}{4pt}}\lambda=d_{\scaleto{\mkern-1.5muH}{4pt}}\eta$.

We now have the following corollaries.

Corollary 5.6**.**

Let $[\omega]\in H^{1,2}({{\mathcal{R}}^{\infty}})$ with unique representative $\omega=dx\wedge\theta^{0}\wedge\epsilon-dt\wedge{\bm{\beta}}(\epsilon)$, $\epsilon=r_{i}\theta^{i}$ as in Theorem 4.1. Then $\Delta$ admits ${\mathcal{E}}=-2r_{i}D_{x}^{i}$ as a variational operator.

Proof.

Starting with equation 5.12, Corollary 4.4 implies

[TABLE]

Equation 5.26 together with the fact ${\bf F}_{Q}=Q_{i}D_{x}^{i}$ gives $\epsilon=\iota^{*}\frac{1}{2}({\bf F}_{Q}-{\bf F}_{Q}^{*})(\vartheta^{0})=r^{i}\theta^{i}$, we have ${\mathcal{E}}={\bf F}_{Q}^{*}-{\bf F}_{Q}=-2r_{i}D_{x}^{i}$ is a variational operator by Theorem 5.4. ∎

Corollary 5.7**.**

Let $[\omega]\in H^{1,2}({{\mathcal{R}}^{\infty}})$ with $\omega=dx\wedge\theta^{0}\wedge(r_{i}\theta^{i})-dt\wedge{\bm{\beta}}(\rho)$ as in Theorem 3.1. Then $\Delta$ admits

[TABLE]

as a variational operator.

Proof.

By Corollary 4.2 the unique representative $\hat{\omega}=dx\wedge\theta^{0}\wedge\epsilon-dt\wedge{\bm{\beta}}(\epsilon)$ has $\epsilon=\frac{1}{2}\left(\rho-\rho^{*}\right)=\frac{1}{2}(r_{i}\theta^{i}-(-X^{i})(r_{i}\theta^{0}))$. Therefore by Corollary 5.6, ${\mathcal{E}}$ in equation 5.27 is a variational operator. ∎

Finally we may also restate Theorem 5.4 without reference to the equation manifold ${{\mathcal{R}}^{\infty}}$ as follows.

Corollary 5.8**.**

The operator ${\mathcal{E}}=r_{i}(t,x,u,u_{x},\ldots)D_{x}^{i}$, $i=0,\ldots,k$ is a variational operator for $u_{t}=K$ if and only if there exists $Q(t,x,u,u_{x},u_{xx},\ldots)$ and $L(t,x,u,u_{x},u_{xx},\ldots)$ such that

[TABLE]

Lastly we combine the results of Theorems 5.1 and 5.4 to prove Theorem 1.1.

Proof.

(Theorem 1.1) Define the linear transformation $\hat{\Phi}:H^{1,2}({{\mathcal{R}}^{\infty}})\to{\mathcal{V}}_{op}(\Delta)$ using Theorem 4.1 by

[TABLE]

where $\epsilon=r_{i}\theta^{i}$ and is skew-adjoint. By Corollary 5.6 $\Phi([\omega])\in{\mathcal{V}}_{op}(\Delta)$.

We check $\hat{\Phi}=\Phi^{-1}$. With ${\mathcal{E}}=-2r_{i}D_{x}^{i}$ a variational operator, let $\epsilon=r_{i}\theta^{i}$ we have from equation 1.4 (or Theorem 5.3) and equation 5.29,

[TABLE]

and

[TABLE]

Therefore $\Phi$ in equation 1.4 is invertible with $\hat{\Phi}$ in equation 5.29 as the inverse. ∎

6. Functional $2$-Forms, Symplectic Forms and Hamiltonian Vector Fields

In this section we quickly review the space of functional forms on $J^{\infty}(\mathbb{R},\mathbb{R})$ as in [anderson:2016a], [anderson:1992a] and relate these to symplectic forms and symplectic operator.

6.1. Functional Forms

On the space $J^{\infty}(\mathbb{R},\mathbb{R})$ with coordinates $(x,u,u_{x},\ldots,u_{i},\ldots)$ the contact forms are $\theta^{i}=du_{i}-u_{i+1}dx$ and $D_{x}=\partial_{x}+u_{x}\partial_{u}+\ldots u_{i+1}\partial_{u_{i}}+\ldots$ is the total $x$ derivative operator. Again $\Omega^{r,s}(J^{\infty}(\mathbb{R},\mathbb{R}))$ denotes the $r$ horizontal, $s$ vertical forms on $J^{\infty}(\mathbb{R},\mathbb{R})$. The horizontal and vertical differentials ${d_{\scaleto{\mkern-1.5muH}{4pt}}}:\Omega^{r,s}(J^{\infty}(\mathbb{R},\mathbb{R}))\to\Omega^{r+1,s}(J^{\infty}(\mathbb{R},\mathbb{R}))$, ${d_{\scaleto{\mkern-1.0muV}{4pt}}}:\Omega^{r,s}(J^{\infty}(\mathbb{R},\mathbb{R}))\to\Omega^{r,s+1}(J^{\infty}(\mathbb{R},\mathbb{R}))$, satisfy

[TABLE]

where $f\in C^{\infty}(J^{\infty}(\mathbb{R},\mathbb{R}))$. Since $d=d_{\scaleto{\mkern-1.5muH}{4pt}}+d_{\scaleto{\mkern-1.0muV}{4pt}}$ this implies,

[TABLE]

The integration by parts operator $I:\Omega^{1,s}(J^{\infty}(\mathbb{R},\mathbb{R}))\to\Omega^{1,s}(J^{\infty}(\mathbb{R},\mathbb{R}))$ is

[TABLE]

and $I$ satisfies the same properties as in 2.3,

[TABLE]

The space of functional $s$ forms $(s\geq 1)$ on $J^{\infty}(\mathbb{R},\mathbb{R})$, ${\mathcal{F}}^{s}(J^{\infty}(\mathbb{R},\mathbb{R}))\subset\Omega^{1,s}(J^{\infty}(\mathbb{R},\mathbb{R}))$, is defined to be the image of $\Omega^{1,s}(E)$ under $I$,

[TABLE]

By definition 6.3, equation 6.1 shows that any $\Sigma\in{\mathcal{F}}^{s}(E)$ can always be written,

[TABLE]

However, not every differential form $\Sigma\in\Omega^{1,s}(J^{\infty}(\mathbb{R},\mathbb{R}))$ written as 6.4 is in the space ${\mathcal{F}}^{s}(J^{\infty}(\mathbb{R},\mathbb{R}))$. In the case of ${\mathcal{F}}^{2}(J^{\infty}(\mathbb{R},\mathbb{R}))$ the following is easy to show using the definition of $I$ in 6.1, see also Proposition 3.7 in [anderson:2016a].

Lemma 6.1**.**

Let $\Sigma\in{\mathcal{F}}^{2}(J^{\infty}(\mathbb{R},\mathbb{R}))$, then there exists a unique skew-adjoint differential operator, ${\mathcal{S}}=s_{i}D_{x}^{i}$ such that,

[TABLE]

The differential $\delta_{\scaleto{\mkern-1.0muV}{4pt}}:{\mathcal{F}}^{s}(J^{\infty}(\mathbb{R},\mathbb{R}))\to{\mathcal{F}}^{s+1}(J^{\infty}(\mathbb{R},\mathbb{R}))$ is defined by

[TABLE]

where we let ${\mathcal{F}}^{0}(J^{\infty}(\mathbb{R},\mathbb{R}))=\Omega^{1,0}(J^{\infty}(\mathbb{R},\mathbb{R}))$. This leads to the differential complex

[TABLE]

which is exact and is known as the Euler complex, see Theorem 2.7 [anderson:1992a].

6.2. Symplectic Forms, Symplectic Operators, and Hamiltonian Vector Fields

Let $\Gamma$ be the Lie algebra of prolonged evolutionary vector fields on $J^{\infty}(\mathbb{R},\mathbb{R})$. We begin by recalling the appropriate definitions (see Section 2.5 [dorfman:1993a]).

Definition 6.2**.**

An element $\Sigma\in{\mathcal{F}}^{2}(J^{\infty}(\mathbb{R},\mathbb{R}))$ is a symplectic form on $\Gamma$ if $\Sigma\neq 0$ and $\delta_{\scaleto{\mkern-1.0muV}{4pt}}(\Sigma)=0$. A skew adjoint differential operator ${\mathcal{S}}=s_{i}D_{x}^{i}$ is symplectic if $dx\wedge\theta^{0}\wedge{\mathcal{S}}(\theta^{0})$ is a symplectic form.

Definition 6.2 combined with Lemma 6.1 shows there is a one-to-one correspondence between symplectic forms and symplectic operators. We now defines Hamiltonian vector fields.

Definition 6.3**.**

Let $\Sigma$ be a symplectic form. A vector field $Y\in\Gamma$ is Hamiltonian if

[TABLE]

Definition 6.3 is equivalent to $\Sigma$ being invariant under the flow of $Y$ on ${\mathcal{F}}^{2}(J^{\infty}(\mathbb{R},\mathbb{R}))$ as shown in the following theorem.

Theorem 6.4**.**

Let $\Sigma$ be a symplectic form. An evolutionary vector field $Y\in\Gamma$ is Hamiltonian with respect to $\Sigma$ if and only if

[TABLE]

where ${\mathcal{L}}^{\natural}=I\circ\pi^{1,2}\circ{\mathcal{L}}$ is the projected Lie derivative on ${\mathcal{F}}^{2}(J^{\infty}(\mathbb{R},\mathbb{R}))$, see Theorem 3.21 in [anderson:2016a].

Proof.

Using Lemma 3.24 in [anderson:2016a] and the fact that $\delta_{\scaleto{\mkern-1.0muV}{4pt}}\Sigma=0$, we have

[TABLE]

By the first property in equation 6.2, $I\circ d_{\scaleto{\mkern-1.0muV}{4pt}}\circ I=I\circ d_{\scaleto{\mkern-1.0muV}{4pt}}$, so conditions 6.7 and 6.8 are equivalent through equation 6.9. ∎

We now write out definition 6.3 in a more familiar form. The exactness of the Euler complex and the condition $\delta_{\scaleto{\mkern-1.0muV}{4pt}}\circ I(Y\mathbin{\raise 0.5pt\hbox{\hbox{{\vbox{\hrule height=0.4pt,width=6.0pt,depth=0.0pt}}}\vrule height=6.0pt,width=0.4pt,depth=0.0pt}\,}\Sigma)=0$ implies there exists $\lambda=2Hdx\in{\mathcal{F}}^{0}(J^{\infty}(\mathbb{R},\mathbb{R}))$ such that

[TABLE]

Writing $Y={\rm pr}(K\partial_{u})$ and $\Sigma=dx\wedge\theta^{0}\wedge{\mathcal{S}}(\theta^{0})$ where ${\mathcal{S}}=s_{i}D_{x}^{i}$ is a skew-adjoint differential operator. The left side of equation 6.10 is then

[TABLE]

Using this computation in 6.10 shows that condition 6.7 (or 6.8) is then equivalent to the following.

Corollary 6.5**.**

Let $\Sigma$ be a symplectic form with corresponding symplectic operator ${\mathcal{S}}$. The evolutionary vector field $Y={\rm pr}(K\partial_{u})\in\Gamma$ is Hamiltonian if and only if there exists $H\in C^{\infty}(J^{\infty}(\mathbb{R},\mathbb{R}))$ such that

[TABLE]

Corollary 6.5 just shows that Definition 6.3 agrees with the standard symplectic Hamiltonian formulation for time independent evolution equations [dorfman:1993a].

6.2.1. Symplectic Potential

If $\Sigma$ is a symplectic form, the exactness of the $\delta_{\scaleto{\mkern-1.0muV}{4pt}}$ complex implies there exists $\psi\in{\mathcal{F}}^{1}(J^{\infty}(\mathbb{R},\mathbb{R}))$ such that $\Sigma=\delta_{\scaleto{\mkern-1.0muV}{4pt}}(\psi)$. The functional form $\psi$ is a symplectic potential for $\Sigma$.

Lemma 6.6**.**

Let $\Sigma\in{\mathcal{F}}^{2}(E)$ be symplectic (and so $\delta_{\scaleto{\mkern-1.0muV}{4pt}}$ closed), then there exists a smooth function $P\in C^{\infty}(J^{\infty}(\mathbb{R},\mathbb{R}))$ such that

[TABLE]

where ${\bf F}_{P}=P_{i}D_{x}^{i}$ is the Fréchet derivative of $P$.

Proof.

A symplectic potential $\psi\in{\mathcal{F}}^{1}(J^{\infty}(\mathbb{R},\mathbb{R}))$ for $\Sigma$ can be written using 6.4 as

[TABLE]

Writing $\Sigma=\delta_{\scaleto{\mkern-1.0muV}{4pt}}\psi$ and using equation 6.14 produces 6.13. ∎

The Hamiltonian condition on $Y\in\Gamma$ in terms of a symplectic potential $\psi$ is the following.

Lemma 6.7**.**

The evolutionary vector field $Y\in\Gamma$ is Hamiltonian for the symplectic form $\Sigma=\delta_{\scaleto{\mkern-1.0muV}{4pt}}\psi$ if and only if there exists $\lambda\in{\mathcal{F}}^{0}(J^{\infty}(\mathbb{R},\mathbb{R}))$ such that

[TABLE]

Proof.

Using the exactness of the $\delta_{\scaleto{\mkern-1.0muV}{4pt}}$ complex we show $\delta_{\scaleto{\mkern-1.0muV}{4pt}}{\mathcal{L}}^{\natural}_{Y}\psi=0$ which is equivalent to equation 6.15. By Theorem 6.4, $Y$ is Hamiltonian if and only if

[TABLE]

where we have used ${\mathcal{L}}^{\natural}_{Y}\circ\delta_{\scaleto{\mkern-1.0muV}{4pt}}=\delta_{\scaleto{\mkern-1.0muV}{4pt}}\circ{\mathcal{L}}^{\natural}_{Y}$ (Lemma 3.24 [anderson:2016a]). This proves the Lemma. ∎

Using either Lemma 6.7 or equations 6.13 and 6.4 we have the following simple corollary.

Corollary 6.8**.**

Let $\Sigma$ be a symplectic form with symplectic potential $\psi=dx\wedge\theta^{0}\cdot P$. The evolutionary vector field $V={\rm pr}(K\partial_{u})\in\Gamma$ is Hamiltonian if and only if there exists $H\in C^{\infty}(J^{\infty}(\mathbb{R},\mathbb{R}))$ such that

[TABLE]

where ${\bf F}_{P}$ is the Fréchet-derivative of $P$ on $J^{\infty}(\mathbb{R},\mathbb{R})$.

A straight forward computation writing $\Sigma=\delta_{V}\psi$ classifies the first order symplectic operators, see also Theorem 6.2 in [dorfman:1993a]

Lemma 6.9**.**

An element $\Sigma\in{\mathcal{F}}^{2}(J^{\infty}(\mathbb{R},\mathbb{R}))$ of the first order form,

[TABLE]

is symplectic, if and only if there exists $P(x,u,u_{x},u_{xx})\in C^{\infty}(J^{\infty}(\mathbb{R},\mathbb{R}))$ depending on up to second order derivative, such that

[TABLE]

6.3. Time Dependent Systems

Most of the definitions and results from Sections 6.1 and 6.2 extend immediately to the case of time dependent systems. Let $E=\mathbb{R}\times J^{\infty}(\mathbb{R},\mathbb{R})$, and label the extra $\mathbb{R}$ with the parameter $t$. The contact forms are

[TABLE]

and we let $\Omega^{r,s}_{\rm t_{sb}}(E)$ be the bicomplex of $t$ semi-basic forms on $E$,

[TABLE]

A generic form $\omega\in\Omega^{1,2}_{\rm t_{sb}}(E)$ is given by

[TABLE]

The anti-derivations $d_{\scaleto{\mkern-1.5muH}{4pt}}^{\scaleto{E}{4pt}}:\Omega^{r,s}_{\rm t_{sb}}(E)\to\Omega^{r+1,s}_{\rm t_{sb}}(E)$ and $d^{\scaleto{E}{4pt}}_{\scaleto{\mkern-1.0muV}{4pt}}:\Omega^{r,s}_{\rm t_{sb}}(E)\to\Omega^{r,s+1}_{\rm t_{sb}}(E)$ are determined by

[TABLE]

and satisfy $(d_{\scaleto{\mkern-1.5muH}{4pt}}^{\scaleto{E}{4pt}})^{2}=0,(d_{\scaleto{\mkern-1.0muV}{4pt}}^{\scaleto{E}{4pt}})^{2}=0,d_{\scaleto{\mkern-1.5muH}{4pt}}^{\scaleto{E}{4pt}}d_{\scaleto{\mkern-1.0muV}{4pt}}^{\scaleto{E}{4pt}}+d_{\scaleto{\mkern-1.0muV}{4pt}}^{\scaleto{E}{4pt}}d_{\scaleto{\mkern-1.5muH}{4pt}}^{\scaleto{E}{4pt}}=0$. However $d\neq d_{\scaleto{\mkern-1.5muH}{4pt}}^{\scaleto{E}{4pt}}+d_{\scaleto{\mkern-1.0muV}{4pt}}^{\scaleto{E}{4pt}}$. The integration by part operator $I$ induces a map $I_{\scaleto{\mkern-2.1muE}{4pt}}:\Omega^{r,s}_{\rm t_{sb}}(E)\to\Omega^{r,s}_{\rm t_{sb}}(E)$ having the formula 6.1 and properties 6.2. We let

[TABLE]

The mapping $\delta_{\scaleto{\mkern-1.0muV}{4pt}}^{\scaleto{E}{4pt}}=I_{\scaleto{E}{4pt}}\circ d_{\scaleto{\mkern-1.0muV}{4pt}}^{\scaleto{E}{4pt}}$ gives rise to the exact sequence as in 6.6. A form $\Sigma\in{\mathcal{F}}_{\rm t_{sb}}^{2}(E)$ is symplectic if $\delta_{\scaleto{\mkern-1.0muV}{4pt}}^{\scaleto{\mkern-2.1muE}{4pt}}\Sigma=0$ and Lemma 6.6 becomes the following.

Lemma 6.10**.**

An element $\Sigma=dx\wedge\theta_{\scaleto{\mkern-2.1muE}{4pt}}^{0}\wedge{\mathcal{S}}(\theta^{0}_{\scaleto{\mkern-2.1muE}{4pt}})\in{\mathcal{F}}_{\rm t_{sb}}^{2}(E)$ where ${\mathcal{S}}=s_{i}D_{x}^{i}$, and $s_{i}\in C^{\infty}(E)$ is symplectic if and only if there exists $P\in C^{\infty}(E)$ such that

[TABLE]

where ${\bf L}_{P}=P_{i}D_{x}^{i}$.

We use Theorem 6.4 to define Hamiltonian vector fields in this case.

Definition 6.11**.**

An evolutionary vector field $Y=pr(K\partial_{u})$ where $K\in C^{\infty}(E)$ is Hamiltonian with respect to the symplectic form $\Sigma\in{\mathcal{F}}^{2}_{\rm t_{sb}}(E)$ if

[TABLE]

where $T=\partial_{t}+Y$ and ${{\mathcal{L}}}^{\natural}_{T}=I_{\scaleto{E}{4pt}}\circ\pi^{1,2}\circ{{\mathcal{L}}}_{T}$ is the projected Lie derivative.

Note that $T$ in Definition 6.11 agrees with $T$ in equation 2.10. We can also write condition 6.23 as follows.

Lemma 6.12**.**

An evolutionary vector field $Y=pr(K\partial_{u})$ is a Hamiltonian vector field for the symplectic form $\Sigma=dx\wedge\theta^{0}_{\scaleto{\mkern-2.1muE}{4pt}}\wedge(s_{i}\theta^{i}_{\scaleto{\mkern-2.1muE}{4pt}})$ if and only if there exists $\xi\in\Omega^{0,2}(E)$ such that

[TABLE]

Proof.

We have ${\rm kernel}\ I_{\scaleto{E}{4pt}}={\rm Image}\ d_{\scaleto{\mkern-1.5muH}{4pt}}^{\scaleto{E}{4pt}}$, therefore equation 6.23 can be written as equation 6.24. ∎

A formula for $\xi$ in equation 6.24 in terms of $\rho=s_{i}\theta^{i}_{\scaleto{\mkern-2.1muE}{4pt}}$ is given in the proof of Theorem 7.5.

The analogue to Lemma 6.7 also holds in this case where $Y$ is replaced by $T$. In order to prove this we now show the commutation formula in equation 6.16 holds where $Y$ is replaced by $T$.

Lemma 6.13**.**

${\mathcal{L}}^{\natural}_{T}\delta_{\scaleto{\mkern-1.0muV}{4pt}}\psi=\delta_{\scaleto{\mkern-1.0muV}{4pt}}{\mathcal{L}}^{\natural}_{T}\psi$.

Proof.

Since $T=\partial_{t}+Y$ and ${\mathcal{L}}^{\natural}_{Y}\delta_{\scaleto{\mkern-1.0muV}{4pt}}\psi=\delta_{\scaleto{\mkern-1.0muV}{4pt}}{\mathcal{L}}^{\natural}_{Y}\psi$ (Lemma 3.24 [anderson:2016a]), we need to check

[TABLE]

We write out both side of this equation. The left side is

[TABLE]

The right side is

[TABLE]

Since the mixed partials are equal $P_{t,i}=P_{i,t}$, equations 6.25 and 6.26 are equal, which proves the Lemma. ∎

Lemma 6.14**.**

The evolutionary vector field $Y\in\Gamma$ is Hamiltonian for the symplectic form $\Sigma=\delta_{\scaleto{\mkern-1.0muV}{4pt}}^{\scaleto{E}{4pt}}\psi$ if and only if there exists $\lambda\in{\mathcal{F}}^{0}(E)$ such that

[TABLE]

Proof.

Using the exactness of the $\delta^{\scaleto{E}{4pt}}_{\scaleto{\mkern-1.0muV}{4pt}}$ complex we show $\delta^{\scaleto{E}{4pt}}_{\scaleto{\mkern-1.0muV}{4pt}}{\mathcal{L}}^{\natural}_{T}\psi=0$ which is equivalent to equation 6.27. By Definition 6.11, $Y$ is Hamiltonian if and only if

[TABLE]

where we have used Lemma 6.13. This proves the Lemma. ∎

Using Lemma 6.14 we have the following corollary which is the $t$-dependent version of Corollary 6.8.

Corollary 6.15**.**

Let $\Sigma=dx\wedge\theta^{0}_{\scaleto{E}{4pt}}\wedge{\mathcal{S}}(\theta^{0}_{\scaleto{E}{4pt}})$ be a symplectic form with symplectic potential $\psi=dx\wedge\theta^{0}\cdot P$. The evolutionary vector field $Y={\rm pr}(K\partial_{u})\in\Gamma$ is Hamiltonian if and only if there exists $H\in C^{\infty}(J^{\infty}(\mathbb{R},\mathbb{R}))$ such that

[TABLE]

Proof.

We just need to compute

[TABLE]

Using this computation in equation 6.27 with $\lambda=2Hdx$ gives equation 6.28. ∎

We call $H$ in 6.28 the Hamiltonian.

Remark 6.16**.**

A symplectic form $\Sigma$ is $t$-invariant if ${\mathcal{L}}_{\partial_{t}}\Sigma=0$. In this case $\Sigma$ determines a well defined symplectic form $\bar{\Sigma}$ on the quotient by the flow of $\partial_{t}$, $q:E\to E/\partial_{t}=J^{\infty}(\mathbb{R},\mathbb{R})$ such that $q^{*}\bar{\Sigma}=\Sigma$. Definition 6.3 where $Y$ and $\Sigma$ are $t$-invariant implies equation 6.11 for $\bar{\Sigma}$ and $\bar{Y}=q_{*}Y$.

7. Variational and Symplectic Operator Equivalence

A time independent evolution equation $u_{t}=K(x,u,u_{x},\dots,u_{n})$ is Hamiltonian [dorfman:1993a] if there exists a symplectic operator ${\mathcal{S}}$ and a function $H$ called the Hamiltonian such that equation 1.5 holds. With this definition, the determination of the symplectic Hamiltonian equations is typically approached in two ways. The first way consists of determining the possible symplectic operators of a certain order [dorfman:1993a]. Then for a given class of symplectic operators ${\mathcal{S}}$, determine $K$ which satisfy equation 1.5. The second approach starts with a given $K$ and then determines if there exists a symplectic operator ${\mathcal{S}}$ such that equation 1.5 holds.

By comparison Theorem 1.3 combines these two questions and resolves the characterization of symplectic Hamiltonian evolution equations by the invariants $H^{1,2}({{\mathcal{R}}^{\infty}})$. This simultaneously solves the existence of ${\mathcal{S}}$ and the existence of the Hamiltonian function $H$ in 1.5.

7.1. $H^{1,2}({{\mathcal{R}}^{\infty}})$ and Symplectic Hamiltonian Evolution Equations.

Given a scalar evolution equation $u_{t}=K(t,x,u,u_{x},\ldots)$, identify the manifolds ${{\mathcal{R}}^{\infty}}$ and $E=\mathbb{R}\times J^{\infty}(\mathbb{R},\mathbb{R})$ by identifying their coordinates which in turn induces an identification of smooth functions. Define the bundle map $\Pi:T^{*}({{\mathcal{R}}^{\infty}})\to T^{*}(E)$ which is a projection operator by

[TABLE]

where $\theta^{i}$ are given in equation 2.11 and $\theta^{i}_{\scaleto{E}{4pt}}$ are in equation 6.19. Also denote by $\Pi$ the induced projection map $\Pi:\Omega^{r,s}({{\mathcal{R}}^{\infty}})\to\Omega_{\rm t_{sb}}^{r,s}(E)$ where for example

[TABLE]

Lemma 7.1**.**

The map $\Pi:\Omega^{r,s}({{\mathcal{R}}^{\infty}})\to\Omega_{\rm t_{sb}}^{r,s}(E)$ is a bicomplex homomorphism,

[TABLE]

Proof.

Equation 7.3 follows for the case $\omega=\theta^{i}$ directly from equations 2.14 and 6.20, and generically from the anti-derivation property of the operators. ∎

Lemma 7.2**.**

The function $\Pi:\Omega^{1,2}({{\mathcal{R}}^{\infty}})\to\Omega_{\rm t_{sb}}^{1,2}(E)$ induces a well defined injective linear map $\widehat{\Pi}:H^{1,2}({{\mathcal{R}}^{\infty}})\to\ker\delta^{\scaleto{E}{4pt}}_{\scaleto{\mkern-1.0muV}{4pt}}\subset{\mathcal{F}}_{\rm t_{sb}}^{i}(E)$ defined by

[TABLE]

where $\omega$ is a representative of $[\omega]$.

Proof.

To show $\widehat{\Pi}$ is well defined, suppose $\omega^{\prime}=\omega+d_{\scaleto{\mkern-1.5muH}{4pt}}\xi$. Then by equation 7.3 and property 3 in equation 6.2 applied to $I_{\scaleto{\mkern-2.1muE}{4pt}}$ gives

[TABLE]

Therefore $\widehat{\Pi}$ is well defined.

We now show $\widehat{\Pi}([\omega])$ is $\delta^{\scaleto{\mkern-1.0muV}{4pt}}_{\scaleto{E}{4pt}}$ closed. We use equation 7.3 and compute

[TABLE]

Since $d_{\scaleto{\mkern-1.0muV}{4pt}}\omega\in H^{1,3}({{\mathcal{R}}^{\infty}})$, Theorem 3.2 implies there exists $\xi\in\Omega^{0,3}$ such that $d_{\scaleto{\mkern-1.0muV}{4pt}}\omega=d_{\scaleto{\mkern-1.5muH}{4pt}}\xi$ so equation 7.5 becomes

[TABLE]

Therefore $\widehat{\Pi}([\omega])$ is $\delta_{\scaleto{\mkern-1.0muV}{4pt}}$ closed.

We now show $\widehat{\Pi}$ is injective. Let $[\omega]\in H^{1,2}({{\mathcal{R}}^{\infty}})$ and let $\omega=dx\wedge\theta^{0}\wedge\epsilon-dt\wedge{\bm{\beta}}(\epsilon)$ be the unique representative from Theorem 4.1, where $\epsilon=r_{i}\theta^{i}$ and $\epsilon^{*}=-\epsilon$. Then $\widehat{\Pi}([\omega])=dx\wedge\theta^{0}_{\scaleto{\mkern-2.1muE}{4pt}}\wedge(s_{i}\theta^{i}_{\scaleto{\mkern-2.1muE}{4pt}})$, and $(s_{i}\theta^{i}_{\scaleto{\mkern-2.1muE}{4pt}})^{*}=-(s_{i}\theta^{i}_{\scaleto{\mkern-2.1muE}{4pt}})$ since $X=D_{x}$. If $\widehat{\Pi}([\omega])=0$, then $s_{i}\theta^{i}_{\scaleto{\mkern-2.1muE}{4pt}}=0$ and $\omega=0$. This shows $\widehat{\Pi}([\omega])\in{\mathcal{F}}_{\rm t_{sb}}^{2}(E)$ and that $\widehat{\Pi}$ is injective. ∎

In particular we have

Corollary 7.3**.**

If $[\omega]\neq 0$ then $\widehat{\Pi}([\omega])\in{\mathcal{F}}_{\rm t_{sb}}^{2}(E)$ is a symplectic form.

We now set out to prove the fact that $\widehat{\Pi}$ in Lemma 7.2 is in fact a bijection which will imply Theorem 1.2 in the Introduction. We will use the following Lemma.

Lemma 7.4**.**

Let $s_{i},\xi_{ij}\in C^{\infty}({{\mathcal{R}}^{\infty}})$ then

[TABLE]

Proof.

Since $dt\wedge\theta^{i}_{\scaleto{\mkern-2.1muE}{4pt}}=dt\wedge\theta^{i}$ and $X=D_{x}$ these identities follow. ∎

We now have the main theorem.

Theorem 7.5**.**

Let ${\mathcal{S}}=s_{i}D_{x}^{i}$ be a skew-adjoint differential operator. The form $\Sigma=dx\wedge\theta^{0}_{\scaleto{\mkern-2.1muE}{4pt}}\wedge(s_{i}\theta^{i}_{\scaleto{\mkern-2.1muE}{4pt}})$ is symplectic, and $Y=pr(K\partial_{u})$ is a Hamiltonian vector-field for $\Sigma$ if and only if

[TABLE]

satisfies $d_{\scaleto{\mkern-1.5muH}{4pt}}\omega=0$, where $\epsilon={\mathcal{S}}(\theta^{0})$.

Proof.

Supposed $\Sigma$ is symplectic and $Y$ is Hamiltonian, then Lemma 6.12 produces $\xi=\xi_{ab}\theta^{a}_{\scaleto{\mkern-2.1muE}{4pt}}\wedge\theta^{b}_{\scaleto{\mkern-2.1muE}{4pt}}$ satisfying equation 6.24. Let

[TABLE]

Equations 7.6 and 6.24 give

[TABLE]

Therefore $[\omega]\in H^{1,2}({{\mathcal{R}}^{\infty}})$. Now Corollary 3.5 implies $\xi_{ab}\theta^{a}\wedge\theta^{b}=-{\bm{\beta}}(s_{i}\theta^{i})$ so that $\omega$ in equation 7.8 and equation 7.7 are the same.

Suppose now that $\omega$ in equation 7.7 is $d_{H}$ closed. By Lemma 7.2, $\Sigma=\widehat{\Pi}(\omega)$ is a symplectic form. So we need only show that $Y$ is Hamiltonian. Again we refer to Lemma 6.12 and show the existence of $\xi=\xi_{ab}\theta^{a}_{\scaleto{\mkern-2.1muE}{4pt}}\wedge\theta^{b}_{\scaleto{\mkern-2.1muE}{4pt}}$ in equation 6.24.

Writing ${\bm{\beta}}(s_{i}\theta^{i})=B_{ab}\theta^{a}\wedge\theta^{b}$ and using equations 7.6 we have

[TABLE]

This will vanish if and only if

[TABLE]

because this term is $t$ semi-basic. Equation 7.10 produces $\xi=-\Pi({\bm{\beta}}(s_{i}\theta^{i}))=B_{ab}\theta^{a}_{\scaleto{\mkern-2.1muE}{4pt}}\wedge\theta^{b}_{\scaleto{\mkern-2.1muE}{4pt}}$ in equation 6.24 and therefore $Y$ is a Hamiltonian vector field for $\Sigma$. ∎

We now summarize the results by the following Theorem whose proof follows directly from Lemma 7.2 and Theorem 7.5

Theorem 7.6**.**

Let $u_{t}=K$ be an evolution equation, and let $Y=pr(K\partial_{u})$ be the evolutionary vector field on $E$ and let ${\mathcal{Z}}_{Y}(E)\subset{\mathcal{F}}_{\rm t_{sb}}^{2}(E)$ be the subset of symplectic forms for which $Y$ is a Hamiltonian vector field. Define the function $\Psi:{\mathcal{Z}}_{Y}(E)\to H^{1,2}({{\mathcal{R}}^{\infty}})$ given by

[TABLE]

where $dx\wedge\theta^{i}_{\scaleto{\mkern-2.1muE}{4pt}}\wedge\epsilon_{\scaleto{\mkern-2.1muE}{4pt}}\in{\mathcal{Z}}_{Y}(E)$ with $\epsilon_{\scaleto{\mkern-2.1muE}{4pt}}={\mathcal{S}}(\theta^{0}_{\scaleto{\mkern-2.1muE}{4pt}})$ and ${\mathcal{S}}=s_{i}D_{x}^{i}$ is the corresponding symplectic operator, and $\epsilon={\mathcal{S}}(\theta^{0})=s_{i}X^{i}(\theta^{0})$. The function $\Psi:{\mathcal{Z}}_{Y}(E)\to H^{1,2}({{\mathcal{R}}^{\infty}})$ is an isomorphism and $\hat{\Psi}=\Psi^{-1}$ where $\hat{\Psi}$ is defined in equation 7.4.

With Theorems 1.1 and 7.6 in hand the proof of Theorem 1.2 and 1.3 are now easily given.

Proof.

(Theorems 1.2 and 1.3) Suppose that ${\mathcal{S}}=s_{i}D_{x}^{i}\in{\mathcal{Z}}_{Y}(E)$ is a symplectic operator for the scalar evolution equation $\Delta=u_{t}-K$ and that $Y={\rm pr}(K\partial_{u})$ is a Hamiltonian vector field for $\Sigma$. Then with $\Psi$ from equation 7.11 in Theorem 7.6 and $\Phi$ in equation 1.4 in Theorem 1.1 we have,

[TABLE]

is a variational operator and so ${\mathcal{S}}$ is a variational operator for $\Delta$ (by the abuse of notation in Remark 2.1). The fact that $\Phi^{-1}\circ\Psi$ is an isomorphism then proves Theorem 1.2.

As above we identify a symplectic operator ${\mathcal{S}}$ on $E$ as an operator on ${J^{\infty}(\mathbb{R}^{2},\mathbb{R})}$, then the function $\Phi$ in equation 1.4 defines an isomorphism between symplectic operators for $\Delta$ and $H^{1,2}({{\mathcal{R}}^{\infty}})$. This proves Theorem 1.3. ∎

As our final Lemma we show for completeness how formula 5.14 can be determined from the symplectic potential.

Lemma 7.7**.**

Let ${\mathcal{S}}=s_{i}D_{x}^{i}$ be a symplectic operator and let $\psi=dx\wedge\theta^{0}_{E}\cdot P\in C^{\infty}(E)$ be a symplectic potential. The unique representative for $\Psi({\mathcal{S}})\in H^{1,2}({{\mathcal{R}}^{\infty}})$ in Theorem 5.3 has $\epsilon={\mathcal{S}}(\theta^{0})$. Furthermore there exists a representative $\omega$ for $\Psi({\mathcal{S}})$ where $\omega$ in equation 5.12 can be written $\omega=d_{\scaleto{\mkern-1.0muV}{4pt}}\eta$ where

[TABLE]

Proof.

By equation 7.11 of Theorem 7.6 we have the unique representative as stated in the Lemma.

To prove the second part of the lemma by using Theorem 4.3 to construct a representative $\omega_{0}$ for $\Psi({\mathcal{S}})$ such that $\omega_{0}=d_{\scaleto{\mkern-1.0muV}{4pt}}\eta_{0}$ with $\eta_{0}=dx\wedge\theta^{0}\cdot Q-dt\wedge\gamma_{0}$.

By equation 4.17 of Corollary 4.4 and equation 6.22 for the operator in the form $\epsilon={\mathcal{S}}(\theta^{0})$ gives

[TABLE]

Lemma 6.10 and equation 7.13 show $\psi_{0}=dx\wedge\theta^{0}_{\scaleto{\mkern-2.1muE}{4pt}}\cdot Q$ is a symplectic potential for ${\mathcal{S}}$ and that $\delta_{\scaleto{\mkern-1.0muV}{4pt}}^{\scaleto{E}{4pt}}\psi_{0}=\delta_{\scaleto{\mkern-1.0muV}{4pt}}^{\scaleto{E}{4pt}}\psi$. Therefore using equation 6.2 (for $I_{\scaleto{E}{4pt}}$) and the exactness of the $d_{\scaleto{\mkern-1.0muV}{4pt}}^{\scaleto{E}{4pt}}$ complex,

[TABLE]

for some $A\in C^{\infty}(E)$ and $\xi\in\Omega^{0,1}(E)$. We then let

[TABLE]

where we are computing $d_{\scaleto{\mkern-1.5muH}{4pt}}$ and $d_{\scaleto{\mkern-1.0muV}{4pt}}$ on ${{\mathcal{R}}^{\infty}}$. Note that by equation 7.3 and 7.14 we have $\Pi(\eta)=\psi$ so that $\eta$ has the form in equation 7.12. We then compute using equation 7.15

[TABLE]

which proves the lemma. ∎

7.2. Time Independent Operators

Equation 1.5 defines when the time independent evolution equation $u_{t}=K(x,u,u_{x},\ldots)$ is a Hamiltonian system with symplectic operator ${\mathcal{S}}$. This is precisely the same definition that the ordinary differential equation $K(x,u,u_{x},\ldots)=0$ admits a variational operator. The following simple lemma is the key to decoupling the variational operator problem for time independent scalar evolution equations.

Lemma 7.8**.**

Let ${\mathcal{S}}=s_{i}D_{x}^{i}$ be a time independent symplectic operator with symplectic potential $P\in C^{\infty}(J^{\infty}(\mathbb{R},\mathbb{R}))$ (equation 6.13 in Lemma 6.6). Then

[TABLE]

Proof.

By the product formula in the calculus of variations (equation 5.80 in [Olver:1993a]) the left side of equation 7.16 is

[TABLE]

Equation 7.17 together with the fact from equation 6.13 $2{\mathcal{S}}(u_{t})={\bf F}_{P}u_{t}-{\bf F}_{P}^{*}u_{t}$ show that the two sides of equation 7.16 agree. ∎

We then have the following.

Theorem 7.9**.**

Let ${\mathcal{S}}$ be a $t$-independent symplectic operator. The following are equivalent,

1. (1)

$u_{t}=K(x,u,u_{x},\ldots)$* is Hamiltonian the sense of ${\mathcal{S}}(K)={\bf E}(H)$.* 2. (2)

${\mathcal{S}}$* is a symplectic variational operator for the ODE $K=0$,* 3. (3)

${\mathcal{S}}$* is a variational operator for $u_{t}=K$ (see Remark 2.1).*

This converts the symplectic Hamiltonian question for the evolution equation into a variational operator problem for the ODE $K=0$, see [fels:2018a].

Proof.

Suppose $u_{t}=K(x,u,...,u_{n})$ is Hamiltonian for the $t$-independent symplectic operator ${\mathcal{S}}$, so that ${\mathcal{S}}(K)={\bf E}(H)$ on $J^{\infty}(\mathbb{R},\mathbb{R})$. Therefore ${\mathcal{S}}$ is a variational operator for the ODE $K=0$. So (1) and (2) are trivially equivalent.

We show (1) implies (3). Suppose that ${\mathcal{S}}(K)={\bf E}(H)$. Using equation 7.17 in Lemma 7.8 we have

[TABLE]

Therefore ${\mathcal{S}}$ is a variational operator for $u_{t}-K$.

Finally we show (3) implies (1). Starting with hypothesis (3) in the form of equation 5.14 we have,

[TABLE]

where $L=L\circ\pi$ (see 2.1). Substituting from equation 7.17 into equation 7.18 we get

[TABLE]

Therefore

[TABLE]

and $u_{t}=K$ is a time independent Hamiltonian evolution equation for the symplectic operator ${\mathcal{S}}$. ∎

8. First Order Operators

For a third order evolution equation

[TABLE]

we write the conditions $\theta^{0}\wedge{\bf L}_{\Delta}^{*}(\epsilon)=0$, when $\epsilon$ is first order and skew-adjoint. This will prove Theorem 1.4 in the Introduction.

Proof.

(Theorem 1.4) By Theorems 5.3 and 4.1 the skew-adjoint operator ${\mathcal{E}}=2RD_{x}+X(R)$ is a variational operator for 8.1 if and only if the skew-adjoint form $\epsilon=-R\theta^{1}-\frac{1}{2}R_{0}\theta^{0}$ is a solution to

[TABLE]

where $K_{i}=\partial_{u_{i}}K$. Using $T(\theta^{0})=d_{\scaleto{\mkern-1.0muV}{4pt}}K=K_{i}\theta^{i}$ and $T(\theta^{1})=X(d_{\scaleto{\mkern-1.0muV}{4pt}}K)=X(K_{i}\theta^{i})$ we have

[TABLE]

The highest possible $\theta^{i}\wedge\theta^{0}$ term in equation 8.2 using 8.3 is $\theta^{4}$. We find from equation 8.2

[TABLE]

While for $\theta^{3}\wedge\theta^{0}$, $\theta^{2}\wedge\theta^{0}$ and $\theta^{1}\wedge\theta^{0}$ we have from equations 8.3 and 8.2,

[TABLE]

For the coefficient of $\theta^{3}\wedge\theta^{0}$ to be zero we have from equation 8.4,

[TABLE]

where $\hat{K}_{2}=\frac{2}{3K_{3}}\left(K_{2}-X(K_{3})\right)$. The coefficient of $\theta^{2}\wedge\theta^{0}$ in equation 8.4 is zero on account of 8.5. For the coefficient of $\theta^{1}\wedge\theta^{0}$ in 8.4 to be zero gives

[TABLE]

Simplifying equation 8.6 using equation 8.5 we get

[TABLE]

It follows that a non-vanishing $R$ (which we may assume to be positive) satisfying equations 8.5 and 8.7 is necessary and sufficient for the existence of a first order variational operator for $\Delta=u_{t}-K$ in equation 8.1 is equivalent to $A=X(\log R)$ and $B=T(\log R)$ satisfying the conditions in Theorem 1.4. This proves Theorem 1.4. ∎

For the KdV equation $u_{t}=u_{xxx}+uu_{x}$ the form $\kappa$ in equation 1.6 is

[TABLE]

Therefore according to Theorem 1.4 there is no first order symplectic formulation for the KdV equation as a Hamiltonian evolution equation.

8.1. First order Hamiltonians Operators and Bi-Hamiltonian Evolution equations

Let $v_{t}={\mathcal{D}}\circ{\bf E}(H(x,v,v_{x},\ldots))$ be a Hamiltonian evolution equation where ${\mathcal{D}}$ is a first order Hamiltonian operator. According to [Olver:1988a] or [vino:1986a] we may choose coordinates (using a contact transformation) such that ${\mathcal{D}}=D_{x}$. The following is Theorem 1 in [Nutku:2002a] in the context of scalar evolution equations.

Lemma 8.1**.**

The potential form of the Hamiltonian evolution equation,

[TABLE]

is given by the equation

[TABLE]

The potential form 8.9 admits ${\mathcal{E}}=D_{x}$ as a first order variational operator, and satisfies

[TABLE]

There is an abuse of notation in this lemma where $D_{x}$ is used as the total $x$ derivative operator in either variable $u$ or $v$ depending on context.

Proof.

Starting with equation 8.8, let $v=u_{x}$ so that 8.8 becomes

[TABLE]

Integrating equation 8.11 with respect to $x$ gives the potential form 8.9.

To prove equation 8.10 holds we simply need the change of variables formula, see exercise 5.49 in [Olver:1993a],

[TABLE]

Equation 8.12 together with the simple fact $-2{\bf E}(u_{t}u_{x})=u_{tx}$ proves equation 8.10. ∎

The second term in the right hand side of equation 8.10 is just the pullback of the Hamiltonian function in 8.8. We also note the following simple corollary.

Corollary 8.2**.**

Every Hamiltonian evolution equation $v_{t}={\mathcal{D}}({\bf E}(H_{1}(x,v,v_{x},\ldots)))$ with first order Hamiltonian operator ${\mathcal{D}}$ is the symmetry reduction of an equation $u_{t}=K(x,u,u_{x},\ldots)$, of the same order, which admits an invariant first order variational operator.

8.2. Bi-Hamiltonian Evolution Equations with a First Order Hamiltonian Operator

We now present sufficient conditions when the potential form of a compatible bi-Hamiltonian system admits a second variational operator.

Theorem 8.3**.**

Let $v_{t}=K(x,v,v_{x},\ldots)=D_{x}\left({\bf E}(H_{1}(x,v,v_{x},\ldots))\right)$ be a Hamiltonian evolution equation with potential form

[TABLE]

Let ${\mathcal{D}}_{0}$ be second time independent Hamiltonian operator with Hamiltonian $H_{0}(x,v,v_{x},\ldots)$ satisfying,

[TABLE]

Assume ${\mathcal{D}}_{0}$ also satisfies the compatibility condition (equation 7.29 in [Olver:1993a])

[TABLE]

Then the right hand side of the potential form satisfies

[TABLE]

where ${\mathcal{E}}={\mathcal{D}}_{0}|_{v=u_{x}}$111 ${\mathcal{D}}_{0}$ is the push-forward of ${\mathcal{E}}$ by the quotient map ${\bf q}:(t,x,u,u_{x},\ldots)\to(t,x,v,v_{x},\ldots)$.. Furthermore if ${\mathcal{E}}={\mathcal{D}}_{0}|_{v=u_{x}}$ is symplectic, then ${\mathcal{E}}$ is a variational operator for the evolution equation 8.13 and

[TABLE]

where $Q$ is defined in equation 5.28 where ${\mathcal{E}}=F_{Q}^{*}-F_{Q}$.

Proof.

First we apply ${\mathcal{E}}={\mathcal{D}}_{0}|_{v=u_{x}}$ to the right hand side of equation 8.13, and use condition 8.14 to get

[TABLE]

Again the last line follows from the change of variables formula in the calculus variations (exercise 5.49 in [Olver:1993a]). This verifies equation 8.15. Then by part (1) of Theorem 7.9 equation 8.15 shows that ${\mathcal{E}}$ is a variational operator for equation 8.13. If $Q$ is the function from equation 5.28 we then have ${\mathcal{E}}(u_{t})=(F_{Q}^{*}-F_{Q})(u_{t})=E(Qu_{t})$ and equation 8.17 that

[TABLE]

∎

Theorem 8.3 makes the hypothesis that ${\mathcal{E}}={\mathcal{D}}|_{v=u_{x}}$ is a symplectic operator. This holds in the case of the Hamiltonian operators given by Theorem 5.3 in [dorfman:1993a],

[TABLE]

satisfy the compatibility conditions with $D_{x}$ in Corollary 3.2 of [cooke:1991a] when $h(v)=(k_{1}v+k_{2})^{-1}$. This gives

[TABLE]

which are symplectic [dorfman:1993a].

9. Examples

Example 9.1**.**

The Harry-Dym equation $z_{t}=z^{3}z_{xxx}$ [dorfman:1993a, wang:2002a] is a compatible bi-Hamiltonian system,

[TABLE]

where

[TABLE]

The change of variable $z=v^{-1}$ maps the Hamiltonian operator $\widehat{\mathcal{D}}_{1}$ to canonical form [vino:1986a, Olver:1988a], and the Hamilonian operators and the associated Hamiltonians in equation 9.2 become

[TABLE]

The Harry-Dym equation 9.1 in these coordinates is then,

[TABLE]

The potential form of equation 9.4 is found by letting $v=w_{x}$ and integrating to get (see also 8.9)

[TABLE]

Equation 8.10 of Lemma 8.1 as it applies to the potential Harry-Dym equation 9.5 produces the following variational operator equation for $D_{x}$,

[TABLE]

We now apply Theorem 8.3 to obtain a second variational operator. The compatibility condition in equation 8.14 is satisfied with the operators from equation 9.3 with

[TABLE]

The operator ${\mathcal{D}}_{0}$ in equation 9.3 is of the form 8.19 so that by equation 8.20

[TABLE]

is a symplectic or variational operator. Since the compatibility condition in equation 8.14 is satisfied and ${\mathcal{E}}$ is a symplectic operator Theorem 8.3 applies. The operator ${\mathcal{E}}$ in equation 9.7 is a variational operator for the potential Harry-Dym equation in 9.5. The function $Q$ in equation 5.28 is easily determined for ${\mathcal{E}}$ (using the fact that $-2{\mathcal{E}}$ is a symplectic operator) to be

[TABLE]

Equation 8.16 with $Q$ in equation 9.8 and $H_{2}$ in equation 9.6 (with $v=w_{x}$) gives the variational operator equation for the potential Harry-Dym equation 9.5,

[TABLE]

If we return to the original coordinates for the Harry-Dym equation and make the change of variable given by $x=u,w=u_{x},w_{x}=u_{xx}u_{x}^{-1},\ldots$ to the potential form in equation 9.5 we get the Schwarzian KdV (or Krichever-Novikov) equation (pg. 120 in [dorfman:1993a]),

[TABLE]

In particular the Schwarzian KdV in equation 9.9 is the potential form of the Harry-Dym equation 9.1. These different coordinate representations of the Harry-Dym and the Schwarzian KdV is summarized by the diagram,

[TABLE]

The variational or symplectic operators for the Schwarzian KdV are obtained by applying the change of variables $x=u,w=u_{x},w_{x}=u_{xx}u_{x}^{-1},\ldots$ to $D_{x}$ and equation 9.7 giving the well known symplectic or variational operators for the Schwarzian KdV [dorfman:1993a],

[TABLE]

With quotient map ${\bf q}(t,x,u,u_{x},u_{xx},\ldots)=(t=t,x=u,z=u_{x},z_{x}=u_{xx}u_{x}^{-1},\ldots)$, the operators from 9.11 project ${\bf q}_{*}{\mathcal{E}}_{i}=\tilde{\mathcal{D}}_{i}$ to the Hamiltonian operators in equation 9.2.

We now compute the explicit unique representative for the $H^{1,2}({{\mathcal{R}}^{\infty}})$ cohomology class for the Schwarzian-KdV 9.9 corresponding to the first operator in 9.11 (Theorem 4.1). This is computed using formula 1.3 in Theorem 1.1 to be,

[TABLE]

We have $d_{\scaleto{\mkern-1.0muV}{4pt}}\omega_{1}=0$ and for the forms $\eta$ and $\lambda$ in Theorem 5.4 we may choose

[TABLE]

Likewise formula 1.3 for the second operator in 9.11 gives the unique cohomology representative ((Theorem 4.1),

[TABLE]

In this case $d_{\scaleto{\mkern-1.0muV}{4pt}}\omega_{0}\neq 0$, but $[\hat{\omega}_{0}]=[\omega_{0}]$ where

[TABLE]

and $d_{\scaleto{\mkern-1.0muV}{4pt}}\omega_{0}=0$. Futhermore with $\omega_{0}$ in equation 9.15 the forms $\eta$ and $\lambda$ in Theorem 5.4 can be chosen to be

[TABLE]

For $\lambda_{i}$ in equations 9.13 and 9.16, it is difficult to determine whether $[\lambda_{i}]\in H^{2,0}({{\mathcal{R}}^{\infty}})$ is trivial or not (see Theorem A.4). However, it is possible but not easy to show $\lambda_{i}\neq d\kappa_{i}$ where $\kappa_{i}$ is $t$-invariant by using the infinite sequence of conservation laws [dorfman:1993a] for the Krichever-Novikov (Schwarzian KdV) equation 9.9. The forms $\lambda_{i}$ define a non-trivial cohomology class in the $t$-invariant variational bi-complex for 9.9.

Example 9.2**.**

The Harry Dym equation can be written in the form

[TABLE]

where the Hamilonian operators and their Hamiltonians are

[TABLE]

Equation 9.17 is obtained from equation 9.4 by substituting $v=-2^{\frac{1}{3}}\sqrt{\hat{v}}$.

Another potential form (or integrable extension) for the Harry-Dym equation 9.17 can be obtained by letting $v=u_{xxx}$ in equation 9.17 so that

[TABLE]

which after integrating three times gives,

[TABLE]

We show that $D_{x}^{3}$ is a variational operator. First using the change of variables formula in the calculus of variation for $v=u_{xxx}$ (exercise 5.49 [Olver:1993a]) we have

[TABLE]

The operator $D_{x}^{3}$ is symplectic which together with equation 9.19 shows that $D_{x}^{3}$ is a variational operator for equation 9.18 and giving,

[TABLE]

In equation 8.14 compatibility was used to show the second Hamiltonian operator for a bi-Hamiltonian equation became a variational operator for the potential form. In order to use a similar argument in this case we need to show ${\mathcal{D}}_{1}{\bf E}(H_{0})={\mathcal{D}}_{0}{\bf E}(H_{-1})$. We find

[TABLE]

In analogy to equation 8.14, this gives rise with $H_{-1}=0$ to the variational operator

[TABLE]

Using the fact that operator ${\mathcal{E}}$ in equation 9.21 is a symplectic operator, the compatibility condition 9.20 gives

[TABLE]

Equation 9.22 shows directly that ${\mathcal{E}}$ in 9.21 is a variational operator for equation 9.18.

It is worth noting that ${\mathcal{E}}(K)=0$ in this example and that $[\omega]=d_{\scaleto{\mkern-1.0muV}{4pt}}[\eta]$ where $[\eta]\in H^{1,1}({{\mathcal{R}}^{\infty}})$. The representative

[TABLE]

with $\epsilon=-\frac{1}{2}{\mathcal{E}}(\theta^{0})=-u_{xxx}\theta^{1}-\frac{1}{2}u_{xxxx}\theta^{0}$ satisfies $\omega=d_{\scaleto{\mkern-1.0muV}{4pt}}\eta$ where (see equation 3.24)

[TABLE]

Since $d_{\scaleto{\mkern-1.5muH}{4pt}}\eta=0$, $[\eta]\in H^{1,1}({{\mathcal{R}}^{\infty}})$. This also produces an example where

[TABLE]

satisfies ${\bf L}_{\Delta}^{*}(Q)=0$, as well as equation A.6. By Theorem A.1, Corollary A.2 or Corollary A.5, $Q$ is not the characteristic of a classical conservation law

Example 9.3**.**

The cylindrical KdV equation is (see [wang:2002a])

[TABLE]

while it’s potential form is

[TABLE]

The form $\kappa$ in equation 1.6 in Theorem 1.4 is $\kappa=t^{-1}dt=d_{\scaleto{\mkern-1.5muH}{4pt}}(\log t)$ and so equation 9.24 admits ${\mathcal{E}}_{1}=tD_{x}$ as a variational operator. In equation 5.14 we have $Q_{1}=-\frac{1}{2}tu_{x}$ leading to

[TABLE]

By solving the equation $\theta^{0}\wedge{\bf L}_{\Delta}^{*}(\epsilon)=0$ from 4.1 for third order forms $\epsilon$ we find that equation 9.24 admits a third order symplectic or variational operator,

[TABLE]

For ${\mathcal{E}}_{0}$ we have

[TABLE]

in equation 5.14 leading to

[TABLE]

If we now compute the reduction of the potential cylindrical KdV by substituting $w=\sqrt{t}\,u_{x}$ into the $x$-derivative of equation 9.24 we get

[TABLE]

where

[TABLE]

Equation 9.25 can of course be obtained from the cylindrical KdV equation 9.23 by the change of variables $w=\sqrt{t}\,v$. It is unclear (to the authors) if the cylindrical KdV in equation 9.24 is a bi-Hamiltonian evolution equation for which ${\mathcal{D}}_{1}$ and ${\mathcal{D}}_{0}$ in equation 9.26 are Hamiltonian operators. Reference [wang:2002a] states there are no Hamiltonians for the cylindrical KdV. It is straightforward to work out the symplectic or variational operators for the potential cylindrical KdV from equation 9.25 following Theorem 8.3.

More generally for any evolution equation of the form

[TABLE]

admits $D_{x}$ as a first variational operator. We find after a long computation that equation 9.27 admits a third order variational in the case where $a(t)\dot{a}(t)\neq 0$ only when

[TABLE]

For the $+$ sign in equation 9.28, the change of variables $t=c_{1}^{-1}(\hat{t}-c_{2}),\ x=c_{1}^{-\frac{1}{3}}\hat{x}$, $u=\frac{1}{2}c_{1}^{\frac{1}{3}}\hat{u}$, takes equation 9.27 with $a(t)$ in 9.28 to the potential form of the cylindrical KdV obtained from equation 9.25. The same result holds in the other cases with slightly different changes of variable.

10. Conclusions

The determination of a symplectic operator for a scalar evolution equation has been shown to be equivalent to the existence of a variational operator which is determined by a non-vanishing cohomology class in $H^{1,2}({{\mathcal{R}}^{\infty}})$. The arguments used to prove this clearly extend to other types of differential equations including systems. In particular Theorem 5.1 holds independently of $\Delta$ being a evolution equation and so the variational operators for $\Delta$ always determine an element of the cohomology $H^{n-1,2}(\Delta)$ as in Theorem 5.1.

There remain many open theoretical questions such as whether the compatibility condition for symplectic operators appears in the cohomology. Another interesting problem is to determine under what conditions the symmetry reduction of a variational operator equation a Hamiltonian system (the converse of Lemma 8.1).

Many difficult computational questions have also not been resolved. We were unable to compute the dimension of $H^{1,2}({{\mathcal{R}}^{\infty}})$ in our examples. Preliminary computations using equation $\theta^{0}\wedge{\bf L}_{\Delta}^{*}(\rho)=0$ from Theorem 3.1 suggests $\dim H^{1,2}({{\mathcal{R}}^{\infty}})=2$ for the Krichever-Novikov equation in Example 1 and others. However we were not able to give a full proof of this fact. We have also not explored in any detail the obvious generalization of Noether’s Theorem which arises from the existence of a variational operator or equivalently by utilizing a non-trivial element of $H^{1,2}({{\mathcal{R}}^{\infty}})$. This would provide an alternate derivation for identifying symmetries and conservation laws for symplectic Hamiltonian systems (see Theorem 7.15 in [Olver:1993a]).

Appendix A The Vertical Differential

The vertical differential induces a mapping $d_{\scaleto{\mkern-1.0muV}{4pt}}:H^{r,s}({{\mathcal{R}}^{\infty}})\to H^{r,s+1}({{\mathcal{R}}^{\infty}})$ defined by $d_{\scaleto{\mkern-1.0muV}{4pt}}[\omega]=[d_{\scaleto{\mkern-1.0muV}{4pt}}\omega]$. Let $u_{t}=K$ be a scalar evolution equation with equation manifold. We now examine when $[\omega]\in{\rm Image}\,d_{\scaleto{\mkern-1.0muV}{4pt}}$.

Theorem A.1**.**

Let $[\zeta]\in H^{1,1}({{\mathcal{R}}^{\infty}})$. There exists $[\kappa]\in H^{1,0}({{\mathcal{R}}^{\infty}})$ such that $[\zeta]=d_{\scaleto{\mkern-1.0muV}{4pt}}[\kappa]$ if and only if $\delta_{\scaleto{\mkern-1.0muV}{4pt}}\circ\Pi(\zeta)=0$ where $\Pi:\Omega^{1,1}({{\mathcal{R}}^{\infty}})\to\Omega^{1,1}_{\rm t_{sb}}(E)$ is defined as in equation 7.2 and $\zeta$ is any representative of $[\zeta]$.

This answers the question of when $[\zeta]$ is the image of a classical conservation law $[\kappa]$. To relate Theorem A.1 to the theory of characteristics for a conservation law, suppose $[\zeta]\in H^{1,1}({{\mathcal{R}}^{\infty}})$ with (unique) canonical representative given in Theorem 3.4 by

[TABLE]

where the function $Q$ satisfies ${\bf L}_{\Delta}^{*}(Q)=0$. Theorem A.1 states that the function $Q$ is the characteristic of a classical conservation law for $\Delta$ if and only if $Q=E(L)$. The test for this condition is the Helmholtz condition $\delta_{\scaleto{\mkern-1.0muV}{4pt}}^{\scaleto{E}{4pt}}(dx\wedge\theta^{0}_{E}\cdot Q)=0$.

Proof.

Suppose $[\zeta]\in H^{1,1}({{\mathcal{R}}^{\infty}})$ where $\zeta=dx\wedge(a_{i}\theta^{i})-dt\wedge\beta$ is a representative, then

[TABLE]

This implies by equation 6.2,

[TABLE]

Let $\mu=m_{i}\theta^{i}\in\Omega^{1,0}({{\mathcal{R}}^{\infty}})$ and

[TABLE]

so that $[\hat{\zeta}]=[\zeta]$. Now by equation 7.3, the definition of $\Pi$ in equation 7.1, and equation A.1,

[TABLE]

Therefore there exists $\hat{\beta}\in\Omega^{0,1}({{\mathcal{R}}^{\infty}})$ such that,

[TABLE]

Now $d_{\scaleto{\mkern-1.5muH}{4pt}}d_{\scaleto{\mkern-1.0muV}{4pt}}\hat{\zeta}=-d_{\scaleto{\mkern-1.0muV}{4pt}}d_{\scaleto{\mkern-1.5muH}{4pt}}\zeta=0$ and therefore from equation A.3,

[TABLE]

However $d_{\scaleto{\mkern-1.0muV}{4pt}}\beta\in\Omega^{0,2}({{\mathcal{R}}^{\infty}})$ and the only way the contact two form $d_{\scaleto{\mkern-1.0muV}{4pt}}\beta$ satisfies equation A.4 is if $d_{\scaleto{\mkern-1.0muV}{4pt}}\beta=0$. This implies from equation A.3 that $d_{\scaleto{\mkern-1.0muV}{4pt}}\hat{\eta}=0$.

Using the vertical exactness of $\Omega^{1,1}({{\mathcal{R}}^{\infty}})$ we conclude there exists $\kappa\in\Omega^{1,0}({{\mathcal{R}}^{\infty}})$ such that $\hat{\zeta}=d_{\scaleto{\mkern-1.0muV}{4pt}}\kappa$. Now

[TABLE]

Again by vertical exactness of the (augmented) variational bicomplex for $d_{\scaleto{\mkern-1.0muV}{4pt}}:\Omega^{2,0}({{\mathcal{R}}^{\infty}})\to\Omega^{2,1}({{\mathcal{R}}^{\infty}})$ applied to $d_{\scaleto{\mkern-1.5muH}{4pt}}\kappa$ we have,

[TABLE]

Since $\mathbb{R}^{2}$ is simply connected we may write

[TABLE]

Finally let

[TABLE]

so that $d_{\scaleto{\mkern-1.0muV}{4pt}}\hat{\kappa}=d_{\scaleto{\mkern-1.0muV}{4pt}}\kappa=\hat{\zeta}$, and equation A.5 gives

[TABLE]

Therefore $[\zeta]=d_{\scaleto{\mkern-1.0muV}{4pt}}[\hat{\kappa}]$ and $[\hat{\kappa}]\in H^{1,0}({{\mathcal{R}}^{\infty}})$. ∎

Corollary A.2**.**

Let $[\zeta]\in H^{1,1}({{\mathcal{R}}^{\infty}})$ with canonical representative given by

[TABLE]

where ${\bf L}_{\Delta}^{*}(Q)=0$ (see Theorem 3.1). Then $[\zeta]=d_{\scaleto{\mkern-1.0muV}{4pt}}[\kappa]$ where $[\kappa]\in H^{1,0}({{\mathcal{R}}^{\infty}})$ if and only if the function $Q$ is in the image of the Euler operator. That is if and only if there exists $A(t,x,u,u_{x},u_{xx},\ldots)\in C^{\infty}(E)$ such that $Q={\bf E}(A)$.

Corollary A.3**.**

If $u_{t}=K$ is even order, then every solution $Q$ to ${\bf L}_{\Delta}^{*}(Q)=0$ is the characteristic of a conservation law.

As is well known, the characteristics of a conservation law are solutions to ${\bf L}_{\Delta}^{*}(Q)=0$ but the converse is not necessarily true and Corollary A.2 identifies those which are. See Example 9.2 for a solution to ${\bf L}_{\Delta}^{*}(Q)=0$ which is not a characteristic of a conservation law.

We now examine the case of $H^{1,2}({{\mathcal{R}}^{\infty}})$.

Theorem A.4**.**

Let $[\omega]\in H^{1,2}({{\mathcal{R}}^{\infty}})$. Then $[\omega]=d_{\scaleto{\mkern-1.0muV}{4pt}}[\eta]$ where $[\eta]\in H^{1,1}({{\mathcal{R}}^{\infty}})$ if and only if $[\omega]\in\ker\Lambda$ where $\Lambda:H^{1,2}({{\mathcal{R}}^{\infty}})\to H^{2,0}({{\mathcal{R}}^{\infty}})$ is defined in equation 4.18.

Proof.

Let $[\omega]\in H^{1,2}({{\mathcal{R}}^{\infty}})$ with representative $\omega$ satisfying $\omega=d_{\scaleto{\mkern-1.0muV}{4pt}}\eta$ and $\lambda$ be as in Lemma 4.5. That is

[TABLE]

Suppose now that $\lambda=d_{\scaleto{\mkern-1.5muH}{4pt}}\kappa$ so that $[\omega]\in\ker\Lambda$. Let $\hat{\eta}=\eta+d_{\scaleto{\mkern-1.0muV}{4pt}}\kappa$. Then

[TABLE]

Therefore $[\omega]=d_{\scaleto{\mkern-1.0muV}{4pt}}[\hat{\eta}]$ where $[\hat{\eta}]\in H^{1,1}({{\mathcal{R}}^{\infty}})$. This proves sufficiency of the condition.

Suppose now that $[\omega]=d_{\scaleto{\mkern-1.0muV}{4pt}}[\eta]$ where $[\eta]\in H^{1,1}({{\mathcal{R}}^{\infty}})$. Let $\omega$ be the representative such that $\omega=d_{\scaleto{\mkern-1.0muV}{4pt}}\eta$. By hypothesis $d_{\scaleto{\mkern-1.5muH}{4pt}}\eta=0$ and so for $\lambda$ in Lemma 4.5 we have

[TABLE]

The same argument as in the second part of the proof of Corollary 4.6 implies that there exists $\kappa\in\Omega^{1,0}(\mathbb{R}^{2})$ such that $\lambda=d_{\scaleto{\mkern-1.5muH}{4pt}}\kappa$. Therefore $\Lambda([\omega])=[\lambda]=[d_{\scaleto{\mkern-1.5muH}{4pt}}\kappa]=0$. ∎

Theorem A.4 is demonstrated in Example 9.2.

As a simple corollary to Theorem A.4 we can identify the elements of $H^{1,1}({{\mathcal{R}}^{\infty}})$ which are not the image of a conservation law as follows.

Corollary A.5**.**

The map $d_{\scaleto{\mkern-1.0muV}{4pt}}:H^{1,1}({{\mathcal{R}}^{\infty}})/d_{\scaleto{\mkern-1.0muV}{4pt}}(H^{1,0}({{\mathcal{R}}^{\infty}}))\to\ker\Lambda$ is an isomorphism. Moreover, we can identify $\eta\in H^{1,1}({{\mathcal{R}}^{\infty}})/d_{\scaleto{\mkern-1.0muV}{4pt}}(H^{1,0}({{\mathcal{R}}^{\infty}}))$ with the space of functions $Q\in C^{\infty}({{\mathcal{R}}^{\infty}})$ such that

[TABLE]