Geometry of B\"acklund Transformations II: Monge-Amp\`ere Invariants

TL;DR

This paper investigates the existence of rank-1 Bäcklund transformations between hyperbolic Euler-Lagrange systems in the plane, identifying obstructions via local invariants and discovering new transformations relating systems of different types.

Contribution

It introduces a framework to determine when Bäcklund transformations exist between hyperbolic Euler-Lagrange systems and uncovers a novel class relating systems of distinct types.

Findings

Identified obstructions to Bäcklund transformations using local invariants.

Established conditions for the existence of rank-1 Bäcklund transformations.

Discovered a new class of transformations connecting systems of different types.

Abstract

This article is concerned with the question: For which pairs of hyperbolic Euler-Lagrange systems in the plane does there exist a rank-$1$ B\"acklund transformation relating them? We express some obstructions to such existence in terms of the local invariants of the Euler-Lagrange systems. In addition, we discover a class of B\"acklund transformations relating two hyperbolic Euler-Lagrange systems of distinct types.

Tables

Table 1. Table 1. Expressions of C j ​ k i subscript superscript 𝐶 𝑖 𝑗 𝑘 C^{i}_{jk} , R k subscript 𝑅 𝑘 R_{k} , S k subscript 𝑆 𝑘 S_{k} , T k subscript 𝑇 𝑘 T_{k} in ( 46 ) and ( 47 ). All C j ​ k i subscript superscript 𝐶 𝑖 𝑗 𝑘 C^{i}_{jk} ( j < k ) 𝑗 𝑘 (j<k) and R k , S k , T k subscript 𝑅 𝑘 subscript 𝑆 𝑘 subscript 𝑇 𝑘 R_{k},S_{k},T_{k} that are not on this list are zero.

$C_{12}^1$	$-\frac{S{R}^2}{T}$	$C_{13}^1$	$\frac{2R^2-1}{2R}$
$C_{14}^1$	$\frac{R^{2}T+2R^2-T}{4S{R}^3}$	$C_{15}^1$	$\frac{4R^2{S}^2+T}{2RT}$
$C_{16}^1$	$-S$	$C_{34}^1$	$R$
$C_{56}^1$	$-1$	$C_{12}^2$	$\frac{4R^4{S}^{2}T+4R^4{S}^2-R^2{T}^2-2R^{2}T+T^2}{4S{R}^{3}T}$
$C_{23}^2$	$\frac{R\left(4R^2{S}^2-3T\right)}{2T}$	$C_{24}^2$	$SR$
$C_{25}^2$	$-\frac{R^2+2}{2R}$	$C_{26}^2$	$\frac{R^{2}T+2R^2-T}{4S{R}^2}$
$C_{34}^2$	$-1$	$C_{56}^2$	$-\frac{1}{R}$
$C_{12}^3$	$-1$	$C_{13}^3$	$\frac{2R^4{S}^{2}T+4R^4{S}^2-2R^2{S}^{2}T-R^2{T}^2-2R^{2}T+T^2}{2S{R}^{3}T}$
$C_{14}^3$	$\frac{1}{R}$	$C_{24}^3$	$-1$
$C_{35}^3$	$\frac{4R^2{S}^2-R^{2}T-T}{2RT}$	$C_{34}^3$	$\frac{4R^4{S}^2-R^{2}T-2R^2+T}{4S{R}^3}$
$C_{45}^3$	$\frac{2S}{R}$	$C_{46}^3$	$-\frac{T}{R^2}$
$C_{36}^3$	$-\frac{4R^2{S}^2-R^{2}T-2R^2+T}{4S{R}^2}$	$C_{13}^4$	$\frac{2\left(2R^2{S}^2-T\right)}{RT}$
$C_{24}^4$	$\frac{S{R}^2}{T}$	$C_{14}^4$	$-\frac{4R^4{S}^2-4R^2{S}^{2}T-R^2{T}^2-2R^{2}T+T^2}{4S{R}^{3}T}$
$C_{34}^4$	$-\frac{1}{2R}$	$C_{36}^4$	$1$
$C_{45}^4$	$-\frac{4R^2{S}^2-T}{2RT}$	$C_{46}^4$	$S$
$C_{56}^4$	$-1$	$C_{15}^5$	$-\frac{2R^4{S}^{2}T+4R^4{S}^2-2R^2{S}^{2}T-R^2{T}^2-2R^{2}T+T^2}{2S{R}^{3}T}$
$C_{12}^5$	$-1$	$C_{16}^5$	$1$
$C_{26}^5$	$R$	$C_{36}^5$	$2S{R}^2$
$C_{46}^5$	$T$	$C_{45}^5$	$-\frac{4R^4{S}^2+R^{2}T+2R^2-T}{4S{R}^3}$
$C_{35}^5$	$-\frac{4R^4{S}^2-R^{2}T-T}{2RT}$	$C_{56}^5$	$-\frac{4R^2{S}^2+R^{2}T+2R^2-T}{4S{R}^2}$
$C_{15}^6$	$-\frac{2\left(2R^2{S}^2-T\right)}{T}$	$C_{16}^6$	$\frac{4R^4{S}^{2}T+4R^4{S}^2-R^2{T}^2-2R^{2}T+T^2}{4S{R}^{3}T}$
$C_{26}^6$	$\frac{S{R}^2}{T}$	$C_{34}^6$	$-1$
$C_{36}^6$	$\frac{R\left(4R^2{S}^2-T\right)}{2T}$	$C_{45}^6$	$-1$
$C_{46}^6$	$SR$	$C_{56}^6$	$-\frac{R}{2}$
$R_3$	$-\frac{4R^4{S}^2-R^{2}T+T}{2T}$	$R_1$	$\frac{2R^4{S}^{2}T+4R^4{S}^2-2R^2{S}^{2}T-R^2{T}^2-2R^{2}T+T^2}{2S{R}^{2}T}$
$R_4$	$-\frac{4R^4{S}^2-R^{2}T-2R^2+T}{4S{R}^2}$	$R_5$	$\frac{4R^2{S}^2+R^{2}T-T}{2T}$
$R_6$	$-\frac{4R^2{S}^2+R^{2}T+2R^2-T}{4SR}$	$S_1$	$\frac{4R^4{S}^{2}T-4R^4{S}^2+8R^2{S}^{2}T-R^2{T}^2+2R^{2}T-3T^2}{4R^{3}T}$
$S_2$	$-\frac{S^2{R}^2}{T}$	$S_3$	$-\frac{SR\left(4R^2{S}^2-T\right)}{2T}$
$S_4$	$-\frac{2R^2{S}^2-T}{2R}$	$S_5$	$-\frac{S\left(8R^2{S}^2+R^{2}T-2T\right)}{2RT}$
$S_6$	$\frac{8R^2{S}^2+R^{2}T+2R^2-3T}{4R^2}$	$T_1$	$\frac{4R^4{S}^{2}T+4R^4{S}^2-R^2{T}^2-2R^{2}T+T^2}{2S{R}^3}$
$T_2$	$-2S{R}^2$	$T_3$	$-\frac{4R^4{S}^2+T}{R}$
$T_4$	$-\frac{T\left(4R^4{S}^2-R^{2}T-2R^2+T\right)}{2S{R}^3}$	$T_5$	$-\frac{T}{R}$
(End of Table 1)

Equations

$$u_{xy}=\frac{1}{2}\sin (2u).$$

$$\left\{\begin{array}[]{l}u_{x}-v_{x}=\lambda\sin(u+v),\\[8.00003pt] u_{y}+v_{y}=\lambda^{-1}\sin(u-v),\end{array}\right.$$

$$v(x,y)=\arctan \left(C\exp \left(-\lambda x-{\lambda }^{-1}y\right)\right),$$

$$A(z_{xx}{z}_{yy}-z_{xy}^2)+B{z}_{xx}+2C{z}_{xy}+D{z}_{yy}+E=0,$$

$$\theta \in \mathcal{I}\cap {\Omega }^1(U)\text{and}\mathrm{d}\theta ,\Omega \in \mathcal{I}\cap {\Omega }^2(U);$$

$$\theta =\mathrm{d}u-p\mathrm{d}x-q\mathrm{d}y,\mathrm{d}\theta =\mathrm{d}x\wedge \mathrm{d}p+\mathrm{d}y\wedge \mathrm{d}q$$

$$\Omega =\left(\mathrm{d}p-\frac{1}{2}\sin (2u)\mathrm{d}y\right)\wedge \mathrm{d}x.$$

$$\mathcal{I}{|}_U=⟨\theta ,{\omega }^1\wedge {\omega }^2,{\omega }^3\wedge {\omega }^4{⟩}_{\mathrm{alg}}.$$

$$\iota :N^6\hookrightarrow M_1\times M_2$$

$${\pi }_1^{\ast }[[\mathrm{d}{\theta }_1,{\Omega }_1]]\equiv {\pi }_2^{\ast }[[\mathrm{d}{\theta }_2,{\Omega }_2]]\mathrm{mod}{\pi }_1^{\ast }{\theta }_1,{\pi }_2^{\ast }{\theta }_2.$$

$$[[{\omega }^1\wedge {\omega }^2,{\omega }^3\wedge {\omega }^4]]\equiv [[{\bar{\omega }}^1\wedge {\bar{\omega }}^2,{\bar{\omega }}^3\wedge {\bar{\omega }}^4]]\mathrm{mod}\theta ,\bar{\theta },$$

$$\begin{array}{rl}[[{\omega }^1\wedge {\omega }^2]]& \equiv [[{\bar{\omega }}^1\wedge {\bar{\omega }}^2]]\mathrm{mod}\theta ,\bar{\theta },\\ [[{\omega }^3\wedge {\omega }^4]]& \equiv [[{\bar{\omega }}^3\wedge {\bar{\omega }}^4]]\mathrm{mod}\theta ,\bar{\theta }.\end{array}$$

$${\bar{\omega }}^i=s^i\theta +t^i\bar{\theta }+u_j^i{\omega }^j$$

$${\bar{\omega }}^i-(s^i\theta +t^i\bar{\theta }+u_j^i{\omega }^j),i=1,\dots ,4.$$

$$\displaystyle\mathrm{d}\left(\begin{array}[]{c}\omega^{0}\\ \omega^{1}\\ \omega^{2}\\ \omega^{3}\\ \omega^{4}\end{array}\right)$$

$$\displaystyle\qquad\qquad\qquad+\left(\begin{array}[]{c}\omega^{1}\wedge\omega^{2}+\omega^{3}\wedge\omega^{4}\\ (V_{1}+V_{5})\omega^{0}\wedge\omega^{3}+(V_{2}+V_{6})\omega^{0}\wedge\omega^{4}\\ (V_{3}+V_{7})\omega^{0}\wedge\omega^{3}+(V_{4}+V_{8})\omega^{0}\wedge\omega^{4}\\ (V_{8}-V_{4})\omega^{0}\wedge\omega^{1}+(V_{2}-V_{6})\omega^{0}\wedge\omega^{2}\\ (V_{3}-V_{7})\omega^{0}\wedge\omega^{1}+(V_{5}-V_{1})\omega^{0}\wedge\omega^{2}\end{array}\right),$$

$$g=\left(\begin{array}[]{ccc}a&{\bf 0}&{\bf 0}\\ {\bf 0}&A&0\\ {\bf 0}&0&B\end{array}\right),~{}A,B\in{\rm GL}(2,\mathbb{R}),~{}a=\det(A)=\det(B),$$

$$J=\left(\begin{array}[]{ccc}1&{\bf 0}&{\bf 0}\\ {\bf 0}&0&I_{2}\\ {\bf 0}&I_{2}&0\end{array}\right)\in{\rm GL}(5,\mathbb{R}).$$

$$S_{1}:=\left(\begin{array}[]{cc}V_{1}&V_{2}\\ V_{3}&V_{4}\end{array}\right),\quad S_{2}:=\left(\begin{array}[]{cc}V_{5}&V_{6}\\ V_{7}&V_{8}\end{array}\right).$$

$$S_i(u\cdot g)=a{A}^{-1}{S}_i(u)B,(i=1,2)$$

$$S_{1}(u\cdot J)=\left(\begin{array}[]{cc}-V_{4}&V_{2}\\ V_{3}&-V_{1}\end{array}\right),\quad S_{2}(u\cdot J)=\left(\begin{array}[]{cc}V_{8}&-V_{6}\\ -V_{7}&V_{5}\end{array}\right).$$

$${\Sigma }_1:=V_3{\omega }^1{\omega }^3-V_1{\omega }^1{\omega }^4+V_4{\omega }^2{\omega }^3-V_2{\omega }^2{\omega }^4$$

$${\Sigma }_2:=V_7{\omega }^1\wedge {\omega }^3-V_5{\omega }^2\wedge {\omega }^3+V_8{\omega }^1\wedge {\omega }^4-V_6{\omega }^2\wedge {\omega }^4$$

$$\mathrm{d}S_{i}\equiv\left(\begin{array}[]{cc}\phi_{4}&-\phi_{2}\\ -\phi_{3}&\phi_{1}\end{array}\right)S_{i}+S_{i}\left(\begin{array}[]{cc}\phi_{5}&\phi_{6}\\ \phi_{7}&\phi_{8}\end{array}\right)\mod\omega^{0},\omega^{1},\ldots,\omega^{4}.$$

$$\mathrm{d}\theta \equiv {\omega }^1\wedge {\omega }^2+{\omega }^3\wedge {\omega }^4\mathrm{mod}\theta .$$

$$\mathrm{d}\theta \equiv \mathrm{d}x\wedge {\hat{\omega }}^2+\mathrm{d}y\wedge {\hat{\omega }}^4\mathrm{mod}\theta ;$$

$$(1/g)\theta =\mathrm{d}z-p\mathrm{d}x-q\mathrm{d}y.$$

$$\mathrm{d}x\wedge \mathrm{d}p+\mathrm{d}y\wedge \mathrm{d}q\equiv (1/g)(\mathrm{d}x\wedge {\hat{\omega }}^2+\mathrm{d}y\wedge {\hat{\omega }}^4)\mathrm{mod}\theta .$$

$${\hat{\omega }}^2\equiv g\mathrm{d}p-gF\mathrm{d}y\mathrm{mod}\mathrm{d}x,\theta ;{\hat{\omega }}^4\equiv g\mathrm{d}q-gF\mathrm{d}x\mathrm{mod}\mathrm{d}y,\theta .$$

$$\mathrm{d}\mathbf{x}={\mathbf{e}}_i{\omega }^i,\mathrm{d}{\mathbf{e}}_j={\mathbf{e}}_i{\omega }_j^i.$$

$$\mathrm{d}{\omega }^i$$

Full text

1. 1 Introduction
2. 2 First Monge-Ampère Invariants
3. 3 The Bäcklund-Pfaff System
4. 4 Obstructions to Integrability
5. 5 A Special Class of Bäcklund Transformations
6. 6 New Examples of Type III
7. 7 Some Classical Examples
8. 8 Open Questions
9. 9 Acknowledgement
10. A Table of Structure Functions
11. B $(M,\mathcal{I}_{0})$ and $(\bar{M},\bar{\mathcal{I}}_{0})$ in PDE Forms
12. C A Note on Calculation

1. Introduction

A Bäcklund transformation is a way to relate solutions of two PDE systems $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$, in such a manner that, given a solution of $\mathcal{E}_{1}$, one can use it to obtain solutions of $\mathcal{E}_{2}$ by solving ODEs, and vice versa. Early studies of Bäcklund transformations date back to the late 19th century.

1.1. Three Examples

The following examples of Bäcklund transformations are classical. 1. Let $u=u(x,y)$ be a harmonic function. One can find a $1$-parameter family of harmonic functions $v=v(x,y)$ by substituing $u(x,y)$ into the Cauchy-Riemann system in $u$ and $v$ and by solving ODEs. In this sense, the Cauchy-Riemann system is a Bäcklund transformation relating solutions of the Laplace equation $\Delta u(x,y)=0$. 2. Let $u=u(x,y)$ be a solution of the sine-Gordon equation

[TABLE]

Substituting $u(x,y)$ in the system (with parameter $\lambda\neq 0$) in $u$ and $v$:

[TABLE]

one obtains a compatible first-order PDE system in $v(x,y)$, which can be solved using ODE methods. In this case, the system (2) is a Bäcklund transformation relating solutions of the sine-Gordon equation. In particular, if we start by setting $u(x,y)=0$, which is a solution of (1), solving (2) (with a fixed $\lambda$) for $v(x,y)$ yields a 1-parameter family of solutions of (1):

[TABLE]

where $C$ is an arbitrary constant. Such $v(x,y)$ are known as $1$-soliton solutions of (1). Iterative application of the Bäcklund transformation (2) yields the so-called $n$-solitons of (1). Interested reader can see [TU00] and [RS02] for more details. 3. Given two immersed surfaces $S_{1},S_{2}\subset\mathbb{E}^{3}$, by a pseudo-spherical (p.s.) line congruence between them we mean immersions $\sigma_{1},\sigma_{2}:U\rightarrow\mathbb{E}^{3}$ ($U\subset\mathbb{R}^{2}$ open) satisfying $\sigma_{i}(U)=S_{i}$ and

1. $(1)$

The distance $d_{\mathbb{E}^{3}}(\sigma_{1}(p),\sigma_{2}(p))$ is a constant $r>0$; 2. $(2)$

For each $p\in U$, the line through $\sigma_{1}(p)$ and $\sigma_{2}(p)$ is tangent to both surfaces at $\sigma_{1}(p)$ and $\sigma_{2}(p)$, respectively; 3. $(3)$

The respective normals $\bm{n}_{1}(p)$ and $\bm{n}_{2}(p)$ form a constant angle $\theta\in(0,\pi)$.

When these conditions hold, $S_{1},S_{2}$ are called the focal surfaces of the corresponding p.s. line congruence.

It is a theorem of Bianchi that $S_{1},S_{2}\subset\mathbb{E}^{3}$ admit a p.s. line conguence (with parameters $r,\theta$) between them only if they both have the Gauss curvature $K=-\sin^{2}\theta/r^{2}$. Conversely, if $S\subset\mathbb{E}^{3}$ satisfies $K=-\sin^{2}\theta/r^{2}$ for some constants $r>0$ and $\theta\in(0,\pi)$, then one can construct a 1-parameter family of p.s. line congruences with parameters $r$ and $\theta$ and with $S$ being a focal surface.

In fact, when $S_{1},S_{2}$ admit a p.s. line congruence with parameters $r$ and $\theta$, the Gauss and Codazzi equations of $S_{i}$ $(i=1,2)$ together form a PDE system $\mathcal{E}_{r,\theta}$. The integrability of $\mathcal{E}_{r,\theta}$ puts conditions on the respective Gauss curvatures. On the other hand, given a surface $S$ with constant Gauss curvature $K=-\sin^{2}\theta/r^{2}<0$, the problem of finding a p.s. line congruence with parameters $r,\theta$ and with $S$ as a focal surface reduces to integrating a Frobenius (aka. completely integrable) system. In this sense, $\mathcal{E}_{r,\theta}$ is a Bäcklund transformation relating surfaces with $K=-\sin^{2}\theta/r^{2}$. For more details, including how $\mathcal{E}_{r,\theta}$ relates to the sine-Gordon equation and its Bäcklund transformations, see [CT80] and [BGG03].

Using the theory of exterior differential systems, one can study Bäcklund transformations from a geometric viewpoint. For instance, the Bäcklund transformation $\mathcal{E}_{r,\theta}$ in Example 3 above is homogeneous in the sense that the symmetry group of the system acts locally transitively on the space of variables. It has rank $1$ in the sense that it generates a 1-parameter family of surfaces with $K=-\sin^{2}\theta/r^{2}$ from a given one. The PDE system for $S$ with a prescribed constant Gauss curvature $K<0$ is an example of a hyperbolic Monge-Ampère system. A complete classification of rank-$1$ homogeneous Bäcklund transformations relating two hyperbolic Monge-Ampère systems has been obtained by Jeanne N. Clelland in [Cle02], using Cartan’s method of equivalence.

1.2. Geometric Formulations

Classically, a Monge-Ampère equation in $z(x,y)$ is a second-order PDE of the form

[TABLE]

where $A,B,C,D,E$ are functions of $x,y,z,z_{x},z_{y}$. This is a class of PDEs that are closely related to surface geometry and calculus of variations (see [Bry99] and [BGG03]). A Monge-Ampère equation (3) is said to be hyperbolic (resp., elliptic, parabolic) if $AE-BD+C^{2}$ is positive (resp., negative, zero).

Geometrically, each PDE system can be formulated as an exterior differential system (see [BCG*+*13]). In particular, since the current work will mainly be concerned with hyperbolic Monge-Ampère systems, we make the definition below, following [BGH95].

Definition 1.1**.**

A hyperbolic Monge-Ampère system is an exterior differential system $(M,\mathcal{I})$, where $M$ is a 5-manifold; $\mathcal{I}\subset\Omega^{*}(M)$ is a differential ideal satisfying: for each $p\in M$, there exists an open neighborhood $U$ of $p$ on which $\mathcal{I}|_{U}$ is algebraically generated by

[TABLE]

these generators must satisfy:

1. (1)

$\theta\wedge(\mathrm{d}\theta)^{2}\neq 0$; 2. (2)

pointwise, ${\rm span}\{\mathrm{d}\theta,\Omega\}$, modulo $\theta$, has rank $2$; 3. (3)

$(\lambda\mathrm{d}\theta+\mu\Omega)^{2}\equiv 0\mod\theta$ has two distinct solutions $[\lambda_{i}:\mu_{i}]\in\mathbb{R}\mathbb{P}^{1}$ $(i=1,2)$.

For example, the sine-Gordon equation (1) may be formulated as a hyperbolic Monge-Ampère system $(M,\mathcal{I})$ in the following way. Let $M\subset\mathbb{R}^{5}$ be an open domain with coordinates $(x,y,u,p,q)$. Let $\mathcal{I}\subset\Omega^{*}(M)$ be the ideal generated by

[TABLE]

and

[TABLE]

It is easy to see that a surface $\iota:S\hookrightarrow M$ satisfying $\iota^{*}(\mathrm{d}x\wedge\mathrm{d}y)\neq 0$ corresponds to a solution of (1) if and only if $\iota^{*}\mathcal{I}=0$. Moreover, the three conditions in Definition 1.1 hold.

In general, let $(M,\mathcal{I})$ be a hyperbolic Monge-Ampère system. It is easy to show that, each point $p\in M$ has an open neighborhood $U$ on which there exist $1$-forms $\theta,\omega^{1},\omega^{2},\omega^{3},\omega^{4}$, everywhere linearly independent, such that111Here $\langle\cdots\rangle_{\rm alg}$ denotes the ideal in $\Omega^{*}(M)$ generated algebraically by the differential forms enclosed in the brackets. Without the subscript ‘alg’, $\langle\cdots\rangle$ denotes the algebraic ideal generated by the differential forms in the bracket and their exterior derivatives.

[TABLE]

Such a list of $1$-forms is called a [math]-adapted (local) coframing of $(M,\mathcal{I})$.

Suppose that $(\theta,\omega^{1},\cdots,\omega^{4})$ is a [math]-adapted local coframing of a hyperbolic Monge-Ampère system $(M,\mathcal{I})$. Let $\iota:S\hookrightarrow M$ be an embedded surface satisfying $\iota^{*}\mathcal{I}=0$ and the independence condition $\iota^{*}(\omega^{1}\wedge\omega^{3})\neq 0$. Such a surface is called an integral manifold (or, in this case, an integral surface) of $(M,\mathcal{I})$. On $S$, the $1$-forms $\omega^{1}$ and $\omega^{2}$ are multiples of each other; so are $\omega^{3}$ and $\omega^{4}$. The equations $\omega^{1}=0$ and $\omega^{3}=0$ each define a tangent line field on $S$. They are just the classical characteristics of the hyperbolic PDE corresponding to $(M,\mathcal{I})$.

Definition 1.2**.**

Given a hyperbolic Monge-Ampère system222The notion of a characteristic system applies to hyperbolic exterior differential systems in general. See [BGH95]. $(M,\mathcal{I})$, where $\mathcal{I}=\langle\theta,\omega^{1}\wedge\omega^{2},\omega^{3}\wedge\omega^{4}\rangle,$ the pair of Pfaffian systems $\mathcal{I}_{10}=\langle\theta,\omega^{1},\omega^{2}\rangle$ and $\mathcal{I}_{01}=\langle\theta,\omega^{3},\omega^{4}\rangle$, which are defined up to ordering, are called the characteristic systems associated to $(M,\mathcal{I})$.

We follow [BGG03] to give the following definition.333 The reader may compare this definition, which suits the current work, with the more general version presented in [Hu19].

Definition 1.3**.**

Given two hyperbolic Monge-Ampère systems $(M_{i},\mathcal{I}_{i})$ $(i=1,2)$ with $\mathcal{I}_{i}=\langle\theta_{i},\mathrm{d}\theta_{i},\Omega_{i}\rangle_{\rm alg},$ a rank-$1$ Bäcklund transformation relating them is a $6$-manifold

[TABLE]

with the natural projections $\pi_{i}:N\rightarrow M_{i}$ satisfying:

1. (1)

Both $\pi_{1}$ and $\pi_{2}$ are submersions, and $\ker(\mathrm{d}\pi_{1})\cap\ker(\mathrm{d}\pi_{2})=0$; 2. (2)

$[\![\pi_{1}^{*}\mathrm{d}\theta_{1},\pi_{2}^{*}\mathrm{d}\theta_{2}]\!]\equiv[\![\pi_{1}^{*}\Omega_{1},\pi_{2}^{*}\Omega_{2}]\!]\mod\pi_{1}^{*}\theta_{1},\pi_{2}^{*}\theta_{2}$.

Here, $[\![\pi_{1}^{*}\mathrm{d}\theta_{1},\pi_{2}^{*}\mathrm{d}\theta_{2}]\!]$ denotes the subbundle of $\Lambda^{2}(T^{*}N)$ generated by $\pi_{1}^{*}\mathrm{d}\theta_{1}$ and $\pi_{2}^{*}\mathrm{d}\theta_{2}$. (The notation $[\![\cdots]\!]$ will be used in a similar way below.) Condition (1) implies that, given any integral surface $S$ of $(M_{1},\mathcal{I}_{1})$, $\pi_{1}^{-1}S$ is $3$-dimensional. Condition (2) implies that, on $N$,

[TABLE]

It follows that, on $\pi_{1}^{-1}S$, $\pi_{2}^{*}\mathcal{I}_{2}$ restricts to be algebraically generated by the single $1$-form $\pi_{2}^{*}\theta_{2}$, so the Frobenius theorem applies. In other words, $\pi_{1}^{-1}S$ is foliated by a 1-parameter family of surfaces whose projections via $\pi_{2}$ are integral surfaces of $(M_{2},\mathcal{I}_{2})$. The same argument works when one starts with an integral surface of $(M_{2},\mathcal{I}_{2})$.

1.3. Obstructions to Existence

Fix two hyperbolic Monge-Ampère systems $(M,\mathcal{I})$ and $(\bar{M},\bar{\mathcal{I}})$, together with a choice of 0-adapted coframings $(\theta,\omega^{1},\cdots,\omega^{4})$ and $(\bar{\theta},\bar{\omega}^{1},\cdots,\bar{\omega}^{4})$ defined on domains $V\subset M$ and $\bar{V}\subset\bar{M}$, respectively. In principle, the problem of finding a rank-$1$ Bäcklund transformation relating solutions of $(M,\mathcal{I})$ with those of $(\bar{M},\bar{\mathcal{I}})$ is a problem of integration. In fact, suppose that $N^{6}\subset V\times\bar{V}$ is a rank-$1$ Bäcklund transformation. The condition (2) in Definition 1.3 implies that, on $N$,

[TABLE]

where pull-back symbols are dropped for clarity.

One can always switch the pairs $(\omega^{1},\omega^{2})$ and $(\omega^{3},\omega^{4})$, if needed, to arrange that

[TABLE]

It follows that, on $N$, there exist functions $s^{i},t^{i},u^{i}_{j}$ $(i,j=1,\ldots,4)$ such that

[TABLE]

with $u^{\alpha}_{r}=u^{s}_{\beta}=0$, $\det(u^{\alpha}_{\beta}),\det(u^{r}_{s})\neq 0$ $(\alpha,\beta=1,2;r,s=3,4)$. Conversely, the existence of a rank-$1$ Bäcklund transformation reduces to analyzing the integrability of a Pfaffian system $(P,\mathcal{J})$, where $P$ is the product $M\times\bar{M}\times U$ with $U$ being a domain for the parameters $s^{i},t^{i}$ and $u^{i}_{j}$; $\mathcal{J}$ is the differential ideal generated by the four $1$-forms

[TABLE]

Theoretically, this is a type of problem that can be handled by the Cartan-Kähler theory (see [BCG*+*13]). However, direct application of the idea above seems to have limited value for two reasons. One, ‘fixing two hyperbolic Monge-Ampère systems’ is not general enough; and the choice of [math]-adapted coframings is quite arbitrary. Two, the calculation involved in analyzing the Pfaffian system $(P,\mathcal{J})$ often quickly becomes enormous and difficult to manage.

On the other hand, we can still analyze the existence of rank-$1$ Bäcklund transformations as an integrability problem. Here we take an analogous but different approach than the one described above. Instead of considering a specific pair of hyperbolic Monge-Ampère systems with chosen adapted coframings, we consider the respective $G$-structure bundles, say, $\mathcal{G}$ and $\bar{\mathcal{G}}$ associated to the Monge-Ampère systems. We then establish a rank-$4$ Pfaffian system $\mathcal{J}^{\prime}$ on $\mathcal{G}\times\bar{\mathcal{G}}\times U^{\prime}$, where $U^{\prime}$ is a parameter space. Suitable integral manifolds of $\mathcal{J}^{\prime}$ correspond to the desired Bäcklund transformations.

This approach, based on the $G$-structure bundles instead of the manifolds of hyperbolic Monge-Ampère systems, allows one to work with all hyperbolic Monge-Ampère systems at the same time. By not specifying a coframing, it is possible to express integrability conditions in terms of the invariants of the Monge-Ampère systems, leading to new ‘obstruction-to-existence’ results.

This addresses the ‘generality issue’ mentioned above. Yet the magnitude of the calculation remains a challenge. Being aware of this, we have assumed, for a significant portion of this work, that the hyperbolic Monge-Ampère systems under consideration are both Euler-Lagrange (Section 4), which is the case for many known examples of Bäcklund transformations. Furthermore, at a certain stage, we assume that the rank-$1$ Bäcklund transformations are of a particular type, which we call ‘special’ (Section 5). Such Bäcklund transformations can be divided into 4 subtypes, which we name as Type I, IIa, IIb and III. Under these assumptions, we obtain our main obstruction results: *Proposition 5.2. $\bf(A)$ If a pair of hyperbolic Euler-Lagrange systems are related by a Type I special Bäcklund transformation, then one of them must be positive, the other negative.

$\bf(B)$ Two hyperbolic Euler-Lagrange systems related by a Type III special Bäcklund transformation cannot be both degenerate.***

Theorem 5.1. *If two hyperbolic Euler-Lagrange systems are related by a Type IIa special rank-$1$ Bäcklund transformation, then each of them corresponds (up to contact equivalence) to a second order PDE of the form $z_{xy}=F(x,y,z,z_{x},z_{y}).$ *

Theorem 5.2. *Let $(M,\mathcal{I})$ and $(\bar{M},\bar{\mathcal{I}})$ be two hyperbolic Euler-Lagrange systems. If $\phi:N\rightarrow M\times\bar{M}$ defines a Type IIb special rank-$1$ Bäcklund transformation relating $(M,\mathcal{I})$ and $(\bar{M},\bar{\mathcal{I}})$, then each of $(M,\mathcal{I})$ and $(\bar{M},\bar{\mathcal{I}})$ must have a characteristic system that contains a rank-1 integrable subsystem. *

In Section 6, we discover examples of Type III special rank-$1$ Bäcklund transformations relating a degenerate hyperbolic Euler-Lagrange system with a non-degenerate one.

We provide a list of open questions in Section 8.

2. First Monge-Ampère Invariants

Let $(M,\mathcal{I})$ be a hyperbolic Monge-Ampère system. Let $\mathcal{G}_{0}$ be the principal bundle over $M$ consisting of [math]-adapted coframes of $(M,\mathcal{I})$. It is easy to verify that $\mathcal{G}_{0}$ is a $G_{0}$-structure, where $G_{0}\subset{\rm GL}(5,\mathbb{R})$ is a $13$-dimensional Lie subgroup. In [BGG03], the reduction of $\mathcal{G}_{0}$ to a $G_{1}$-structure $\mathcal{G}_{1}$ is performed such that the following structure equations hold on $\mathcal{G}_{1}$:

[TABLE]

where $\phi_{0}=\phi_{1}+\phi_{4}=\phi_{5}+\phi_{8}$, and $G_{1}\subset G_{0}$ is the subgroup generated by

[TABLE]

and

[TABLE]

Definition 2.1**.**

Let $(M,\mathcal{I})$ be a hyperbolic Monge-Ampère system. A $1$-adapted coframing of $(M,\mathcal{I})$ with domain $U\subset M$ is a section $\bm{\eta}:U\rightarrow\mathcal{G}_{1}$.

Following [BGG03], we introduce the notation444To be precise, these $S_{i}$ are $1/2$ times those defined in [BGG03] with the same notation.

[TABLE]

It is shown in [BGG03] that

Proposition 2.1**.**

Along each fiber of $\mathcal{G}_{1}$,

[TABLE]

for any $g={\rm diag}(a;A;B)$ in the identity component of $G_{1}$. Moreover,

[TABLE]

Proposition 2.1 has a simple interpretation: the matrices $S_{1}$ and $S_{2}$ correspond to two invariant tensors under the $G_{1}$-action. More explicitly, one can verify that the quadratic form

[TABLE]

and the $2$-form

[TABLE]

are $G_{1}$-invariant, which implies that $\Sigma_{1},\Sigma_{2}$ are locally well-defined on $M$.

An infinitesimal version of Proposition 2.1 will be useful: for $i=1,2$,

[TABLE]

An Euler-Lagrange system, in the classical calculus of variations, is a system of PDEs whose solutions correspond to the stationary points of a given first-order functional. In [BGG03], an Euler-Lagrange system is formulated as a Monge-Ampère system555See Definitions 1.3 and 1.4 of [BGG03]; moreover, it is shown:

Proposition 2.2**.**

([BGG03])* A hyperbolic Monge-Ampère system is locally equivalent to an Euler-Lagrange system if and only if $S_{2}$ vanishes.*

Remark 1*.*

Proposition 2.2 says that the property of being Euler-Lagrange is intrinsic. From now on, we will treat this Proposition as our ‘definition’ of hyperbolic Euler-Lagrange (Monge-Ampère) systems.

Proposition 2.3**.**

([BGG03])* A hyperbolic Monge-Ampère system corresponds to the wave equation $z_{xy}=0$ (up to contact equivalence) if and only if $S_{1}=S_{2}=\bm{0}$.*

The following result will also be useful.

Proposition 2.4**.**

A hyperbolic Monge-Ampère system $(M,\mathcal{I})$, where $\mathcal{I}$ is algebraically generated by $\theta$, $\omega^{1}\wedge\omega^{2}$ and $\omega^{3}\wedge\omega^{4}$, locally corresponds to a PDE of the form $z_{xy}=F(x,y,z,z_{x},z_{y})$ (up to contact equivalence) if and only if each of the characteristic systems $\mathcal{I}_{10}=\langle\theta,\omega^{1},\omega^{2}\rangle$ and $\mathcal{I}_{01}=\langle\theta,\omega^{3},\omega^{4}\rangle$ admits a rank-$1$ integrable subsystem.

Proof. One direction is immediate. In fact, formulating the PDE $z_{xy}=F(x,y,z,z_{x},z_{y})$ as a hyperbolic Monge-Ampère system, one easily notices that $\mathrm{d}x$ and $\mathrm{d}y$, respectively belonging to the two characteristic systems, are integrable.

For the other direction, assume that $(M,\mathcal{I})$ has the property that each of $\mathcal{I}_{10}$ and $\mathcal{I}_{01}$ has a rank-$1$ integrable subsystem; and let $(\theta,\omega^{1},\ldots,\omega^{4})$ be a coframing defined on a domain $U\subset M$ satisfying

[TABLE]

For $\mathcal{I}_{10}$, this means that a certain linear combination $A\theta+B\omega^{1}+C\omega^{2}$, where $A,B,C$ (not all zero) are functions on $U$, is closed; hence, by shrinking $U$, if needed, we can find a function $x$ defined on $U$ so that $\mathrm{d}x=A\theta+B\omega^{1}+C\omega^{2}$. Since $\theta$ is a contact form, $B,C$ cannot both be zero. Without loss of generality, assume that $B\neq 0$. Let $\hat{\omega}^{1}=A\theta+B\omega^{1}+C\omega^{2}=\mathrm{d}x$ and $\hat{\omega}^{2}=(1/B)\omega^{2}$. Similarly, there exist functions $A^{\prime},B^{\prime},C^{\prime}$ (assuming $B^{\prime}\neq 0$) and $y$ such that $\hat{\omega}^{3}=A^{\prime}\theta+B^{\prime}\omega^{3}+C^{\prime}\omega^{4}=\mathrm{d}y$. Let $\hat{\omega}^{4}=(1/{B^{\prime}})\omega^{4}$.

Now we have

[TABLE]

hence, the system $\langle\theta,\mathrm{d}x,\mathrm{d}y\rangle$ is completely integrable. By the Frobenius theorem, there exists a function $z$ such that $\langle\theta,\mathrm{d}x,\mathrm{d}y\rangle=\langle dz,\mathrm{d}x,\mathrm{d}y\rangle$. In other words, there exist functions $g,p,q$ $(g\neq 0)$ defined on $U$ such that

[TABLE]

This implies that

[TABLE]

By Cartan’s Lemma, there exists a function $F$ such that

[TABLE]

The vanishing of $\theta$ and $\hat{\omega}^{1}\wedge\hat{\omega}^{2}$ on integral surfaces then implies that locally the corresponding Monge-Ampère equation is equivalent to $z_{xy}=F(x,y,z,z_{x},z_{y})$. ∎Now we turn to hyperbolic Euler-Lagrange systems.

By Proposition 2.1, the sign of $\det(S_{1})$ is independent of the choice of 1-adapted coframings. Hence, each hyperbolic Euler-Lagrange system belongs to exactly one of the following three classes.

Definition 2.2**.**

Given a hyperbolic Euler-Lagrange system $(M,\mathcal{I})$, it is said to be

• •

positive if $\det(S_{1})>0$;

• •

negative if $\det(S_{1})<0$;

• •

degenerate if $\det(S_{1})=0$.

Example 1**.**

The oriented orthonormal frame bundle $\mathcal{O}$ over the Euclidean space $\mathbb{E}^{3}$ consists of elements of the form $({\bf x},{\bf e}_{1},{\bf e}_{2},{\bf e}_{3})$, where ${\bf x}\in\mathbb{E}^{3}$, and $({\bf e}_{1},{\bf e}_{2},{\bf e}_{3})$ is an oriented orthonormal frame at ${\bf x}$. On $\mathcal{O}$, define the $1$-forms $\omega^{i}$ and $\omega^{i}_{j}$ by

[TABLE]

We have the standard structure equations:

[TABLE]

where $\omega^{i}_{j}+\omega^{j}_{i}=0$. Consider the natural quotient

[TABLE]

The differential forms $\omega^{3},\mathrm{d}\omega^{3},\mathrm{d}\omega^{1}_{2}+\omega^{1}\wedge\omega^{2}$ are annihilated by $\ker(\mathrm{d}\pi)$ and are invariant along the fibres of $\pi$. Therefore, they are defined on $M$. The exterior differential system $(M,\mathcal{J})$, where

[TABLE]

is hyperbolic Monge-Ampère. Its integral surfaces correspond to generalized surfaces in $\mathbb{E}^{3}$ with Gauss curvature $K=-1$.

Now consider the change of basis

[TABLE]

[TABLE]

In terms of the $\eta^{i}$ $(i=0,\ldots,4)$, we have $\mathcal{J}=\langle\eta^{0},\eta^{1}\wedge\eta^{2},\eta^{3}\wedge\eta^{4}\rangle_{\rm alg}$. It can be verified that $\bm{\eta}:=(\eta^{0},\eta^{1},\ldots,\eta^{4})$ is a local $1$-adapted coframing of $(M,\mathcal{J})$. Calculating using this coframing, we obtain $V_{2}=V_{3}=1$ with all other $V_{i}$ being zero. It follows that $(M,\mathcal{J})$ is a hyperbolic Euler-Lagrange system of the negative type. (For a presentation using coordinates, see Appendix C.)

Example 2**.**

Consider a PDE of the form $z_{xy}=f(z)$. (This is called an $f$-Gordon equation.) It corresponds to a hyperbolic Euler-Lagrange system of the degenerate type, for it is easy to verify that

[TABLE]

[TABLE]

form a 1-adapted coframing of the corresponding hyperbolic Monge-Ampère system. Using this coframing, one can calculate that $V_{3}=-f^{\prime}(z)$ with all other $V_{i}$ being identically zero.

Example 3**.**

Consider the Lorentzian space $\mathbb{E}^{2,1}$. Let $\pi:\mathcal{F}\rightarrow\mathbb{E}^{2,1}$ be the oriented pseudo-orthonormal frame bundle (with ${\rm SO}(2,1)$ fibres) consisting of $({\bf x},{\bf e}_{1},{\bf e}_{2},{\bf e}_{3})$ satisfying

[TABLE]

where $\cdot$ stands for the inner product on $\mathbb{E}^{2,1}$ with the signature $(+,-,+)$.

Define the $1$-forms $\omega^{i},\omega^{i}_{j}$ on $\mathcal{F}$ by

[TABLE]

We have the structure equations

[TABLE]

with $\omega^{2}_{\alpha}=\omega^{\alpha}_{2}$  $(\alpha=1,3)$ and $\omega^{1}_{3}+\omega^{3}_{1}=0$. Consider the quotient

[TABLE]

where $\mathcal{H}$ is the hyperboloid in $\mathbb{R}^{3}$ defined by $x^{2}-y^{2}+z^{2}=1$.

The differential forms $\omega^{3},\mathrm{d}\omega^{3},\omega^{1}\wedge\omega^{2}-\omega^{1}_{3}\wedge\omega^{2}_{3}$ are annihilated by $\ker(\mathrm{d}\pi)$ and are invariant along the fibres of $\pi$. Thus they are defined on $M$. The exterior differential system $(M,\mathcal{J})$, where

[TABLE]

is hyperbolic Monge-Ampère. Its integral surfaces correspond to time-like (since ${\bf e}_{3}$, being space-like, is the normal) surfaces in $\mathbb{E}^{2,1}$ with the constant Gauss curvature $K=1$.

Under the change of basis

[TABLE]

[TABLE]

one can verify that $\bm{\eta}:=(\eta^{0},\eta^{1},\ldots,\eta^{4})$ is a local $1$-adapted coframing of $(M,\mathcal{J})$. Computing using this coframing, we find that $V_{2}=1$, $V_{3}=-1$, all other $V_{i}$ being zero. It follows that $(M,\mathcal{J})$ is a hyperbolic Euler-Lagrange system of the positive type. (For a presentation using coordinates, see Appendix C.)

Remark 2*.*

One can verify that the hyperbolic Monge-Ampère systems occurring in Clelland’s classification [Cle02] of homogeneous rank-$1$ Bäcklund transformations are Euler-Lagrange.

Proposition 2.5**.**

A hyperbolic Euler-Lagrange system of either the positive or the negative type is not contact equivalent to any PDE of the form $z_{xy}=F(x,y,z,z_{x},z_{y})$.

Proof. Given a hyperbolic Monge-Ampère system $(M,\mathcal{I})$, if $S_{2}=0$ and $S_{1}$ nonsingular, we can compute the derived systems666Given a Pfaffian system $I\subset T^{*}M$, its derived systems are defined recursively: $I^{(0)}=I$, $I^{(j+1)}=[\![\omega\in\Gamma(I^{(j)})|~{}\mathrm{d}\omega\equiv 0\mod I^{(j)}]\!]\subset T^{*}M$. There always exists a $k\geq 0$ such that $I^{(\ell)}=I^{(k)}=:I^{(\infty)}$ for all $\ell\geq k$. Any integrable subsystem of $I$ is contained in $I^{(\infty)}$. (see [BCG*+*13]) of $I_{10}=[\![\omega^{0},\omega^{1},\omega^{2}]\!]$ using (19) and find

[TABLE]

It follows that $I_{10}$ has no integrable subsystem. A similar argument works for $I_{01}$. By Proposition 2.4, $(M,\mathcal{I})$ cannot be contact equivalent to a PDE of the form $z_{xy}=F(x,y,z,z_{x},z_{y})$.∎

3. The Bäcklund-Pfaff System

In this section, we prove that, given two hyperbolic Monge-Ampère systems, the existence of a rank-$1$ Bäcklund transformation relating them can be formulated as the integrability of a rank-$4$ Pfaffian system.

We start by fixing some notations.

Let $(M,\mathcal{I})$ and $(\bar{M},\bar{\mathcal{I}})$ be two hyperbolic Monge-Ampère systems. Let ${\mathcal{G}}_{1}$ and $\bar{\mathcal{G}}_{1}$ be the respective $G_{1}$-structures (see Section 2). Let $\alpha=(\alpha_{0},\alpha_{1},\ldots,\alpha_{4})$ and $\beta=(\beta_{0},\beta_{1},\ldots,\beta_{4})$ be the tautological 1-forms on $\mathcal{G}_{1}$ and $\bar{\mathcal{G}}_{1}$, respectively. And let $\mathcal{P}$ be the product of $\mathcal{G}_{1}\times\bar{\mathcal{G}_{1}}$ with a space of parameters:

[TABLE]

Proposition 3.1**.**

An embedded $6$-manifold $\phi:N\rightarrow M\times\bar{M}$ is a rank-$1$ Bäcklund transformation if and only if $N$ admits a lifting $\hat{\phi}:N\rightarrow\mathcal{P}$ that is an integral manifold of a rank-$4$ Pfaffian system $\mathcal{J}$ with the independence condition $\alpha_{1}\wedge\alpha_{2}\wedge\alpha_{3}\wedge\alpha_{4}\wedge\alpha_{0}\wedge\beta_{0}\neq 0$, where $\mathcal{J}$ is generated by

[TABLE]

at each $(u,\bar{u},s_{1},\ldots,s_{4},t_{1},\ldots,t_{4},\mu,\epsilon)\in\mathcal{P}$.

Proof. Let $\tau,\pi,\bar{\pi}$ denote the obvious projections (see the diagram above). By the construction of $\mathcal{G}_{1}$ and $\bar{\mathcal{G}}_{1}$, on $\mathcal{P}$, we have

[TABLE]

and similarly for $\bar{\mathcal{I}}$ and the $\beta_{i}$. Now assume that $\phi$ admits a lifting $\hat{\phi}$ that integrates the Pfaffian system $\mathcal{J}$. It is easy to see that, on $N$,

[TABLE]

In the last congruence, we have used the assumption that $\mu^{2}\neq\epsilon$, which guarantees that the bundle $[\![\alpha_{1}\wedge\alpha_{2}+\alpha_{3}\wedge\alpha_{4},~{}\beta_{1}\wedge\beta_{2}+\beta_{3}\wedge\beta_{4}]\!]$ has rank $2$ modulo $\alpha_{0},\beta_{0}$ when pulled back to $N$. It follows that $\phi:N\rightarrow M\times\bar{M}$ defines a rank-$1$ Bäcklund transformation.

Conversely, suppose that $\phi:N\rightarrow M\times\bar{M}$ defines a rank-$1$ Bäcklund transformation. Let $\bm{\eta}:=(\eta_{0},\eta_{1},\ldots,\eta_{4})$ (resp. $\bm{\xi}:=(\xi_{0},\xi_{1},\ldots,\xi_{4})$) be a local 1-adapted coframing defined on a domain in $M$ (resp. $\bar{M}$). We have, by definition,

[TABLE]

By switching the pairs $(\eta_{1},\eta_{2})$ and $(\eta_{3},\eta_{4})$ and by shrinking $N$, if needed, we can assume that, on $N$,

[TABLE]

where the pull-back symbol is dropped for convenience. Consequently, there exist 16 functions $s_{1},\ldots,s_{4},t_{1},\ldots,t_{4},u_{1},\ldots,u_{8}$ defined on $N$ such that, when restricted to $N$,

[TABLE]

Here $u_{1}u_{4}-u_{2}u_{3}$ and $u_{5}u_{8}-u_{7}u_{6}$ are both nonvanishing. Moreover, since $[\![\mathrm{d}\eta_{0},\mathrm{d}\xi_{0}]\!]$ has rank $2$ modulo $\eta_{0},\xi_{0}$, we have

[TABLE]

Using the flexibility of choosing the $1$-adapted coframings $\bm{\eta}$ and $\bm{\xi}$ (see (26), (27)), we can normalize some of the $u_{i}$. To be specific, we can apply ${\rm SL}(2,\mathbb{R})$-actions (i.e., acting on $\bm{\eta}$ pointwise by a matrix $g$ in the form of (26) with $a=1$) to $(\eta_{1},\eta_{2})$ and $(\eta_{3},\eta_{4})$ to arrange that

[TABLE]

and

[TABLE]

Then we transform $(\eta_{0},\eta_{1},\ldots,\eta_{4})$ pointwise by a diagonal matrix in $G_{1}$ to arrange that

[TABLE]

Meanwhile, this transformation scales the $s_{i}$ $(i=1,\ldots,4)$. Finally, if $u_{1}=u_{4}<1$, then we switch the pairs of indices $(1,2)$ and $(3,4)$ for $\bm{\xi}$ and $\bm{\eta}$ in $\eqref{xieta}$ and multiply the new $\eta_{0},\eta_{2},\eta_{4}$ by $\epsilon$; the resulting $u_{i}$ will satisfy

[TABLE]

It follows that the $(\bm{\eta},\bm{\xi},(s_{i}),(t_{i}),u_{1},\epsilon)$ thus constructed defines a map $\hat{\phi}:N\rightarrow\mathcal{P}$ that is a lifting of $\phi$ and an integral manifold of $\mathcal{J}$. ∎

In light of Proposition 3.1, we make the following definitions.

Definition 3.1**.**

The system $(\mathcal{P},\mathcal{J})$ in Proposition 3.1 is called the [math]-refined Bäcklund-Pfaff system for rank-$1$ Bäcklund transformations relating two hyperbolic Monge-Ampère systems.

Definition 3.2**.**

A $6$-dimensional integral manifold (satisfying the independence condition described in Proposition 3.1) of the [math]-refined Bäcklund-Pfaff system $(\mathcal{P},\mathcal{J})$ is called a [math]-refined lifting of the underlying Bäcklund transformation.

In these terms, Proposition 3.1 says that each rank-$1$ Bäcklund transformation relating two hyperbolic Monge-Ampère systems has a [math]-refined lifting. Of course, given such a Bäcklund transformation $\phi:N\rightarrow M\times\bar{M}$, its [math]-refined liftings are not unique.

Lemma 3.1**.**

Let $\phi:N\rightarrow M\times\bar{M}$ be a rank-$1$ Bäcklund transformation relating two hyperbolic Monge-Ampère systems. The functions $\mu\circ\hat{\phi}$ and $\epsilon\circ\hat{\phi}$ are independent of the choice of [math]-refined liftings $\hat{\phi}$ of $\phi$.

Proof. Clearly, for different choices of $\hat{\phi}$, the $1$-forms $\phi^{*}\alpha_{0}$ and $\phi^{*}\beta_{0}$ only change by scaling. On $N$, the quotient $\lambda_{1}/\lambda_{2}$ between the two solutions $\lambda_{1},\lambda_{2}$ of the equation

[TABLE]

is independent of the scaling of $\alpha_{0}$ and $\beta_{0}$ but may depend on the order of the pair $(\lambda_{1},\lambda_{2})$. By (37), either $\lambda_{1}/\lambda_{2}$ or $\lambda_{2}/\lambda_{1}$ must be equal to $\epsilon\mu^{4}$.

When $|\lambda_{1}/\lambda_{2}|=1$, it is necessary that $\mu=1$ and $\epsilon=-1$. When $|\lambda_{1}/\lambda_{2}|\neq 1$, the ambiguity of ordering $\lambda_{1}$ and $\lambda_{2}$ can be eliminated by requiring that $|\lambda_{1}/\lambda_{2}|>1$, and hence $\lambda_{1}/\lambda_{2}=\epsilon\mu^{4}$. It follows that $\mu$ and $\epsilon$ are both independent of the lifting.∎

Remark 3*.*

A. There is a simple geometric interpretation for the two possible values of $\epsilon$. Let $(M,\mathcal{I})$ and $(\bar{M},\bar{\mathcal{I}})$ be as above. Suppose that $\phi:N\rightarrow M\times\bar{M}$ defines a rank-$1$ Bäcklund transformation. The 4-plane field $\mathcal{D}:=[\![\alpha_{0},\beta_{0}]\!]^{\perp}$ on $N$ is independent of the choice of [math]-refined liftings of $\phi$. When restricted to $\mathcal{D}$, the $4$-forms $(\mathrm{d}\alpha_{0})^{2}$ and $(\mathrm{d}\beta_{0})^{2}$ define two orientations. If $\epsilon=1$, these two orientations are the same; if $\epsilon=-1$, they are distinct.

B. If, for a rank-$1$ Bäcklund transformation relating two hyperbolic Monge-Ampère systems, $\epsilon=1$, then $\mu>1$. This is because, if $\mu=1$, then the condition (2) in Definition 1.3 will not hold.

4. Obstructions to Integrability

In this section, we express some obstructions to the integrability of $(\mathcal{P},\mathcal{J})$ in terms of the invariants of the two hyperbolic Monge-Ampère systems.

For convenience, we introduce new notations below.

1. (A)

Let $\eta_{1}:=\alpha_{1},~{}\eta_{2}:=\alpha_{2},~{}\eta_{3}:=\alpha_{3},~{}\eta_{4}:=\alpha_{4},~{}\eta_{5}:=\alpha_{0},~{}\eta_{6}:=\beta_{0}.$ 2. (B)

The components of the pseudo-connection 1-form on $\mathcal{G}_{1}$ $($see (19)$)$ are denoted by $\varphi_{0},\varphi_{1},\ldots,\varphi_{8}$; those on $\bar{\mathcal{G}}_{1}$ are denoted by $\psi_{0},\psi_{1},\ldots,\psi_{8}$.

On $\mathcal{P}$, differentiating the $\theta_{i}$ and reducing modulo $\theta_{1},\ldots,\theta_{4}$ yields the following congruences:

[TABLE]

where summation over repeated indices is intended; $\pi_{i\alpha}$ $(i=1,\ldots,4;\alpha=1,\ldots,6)$ are linear combinations of the 1-forms in the set

[TABLE]

the components of the torsion $T:=(\tau_{i})$ are of the form

[TABLE]

for some functions $T_{ijk}$ defined on $\mathcal{P}$ satisfying $T_{ijk}=-T_{ikj}$.

We use a standard method (see [BCG*+*13]) to obtain from (39) obstructions to the integrability of $(\mathcal{P},\mathcal{J})$. The key is that, on an integral manifold of $(P,\mathcal{J})$, $\theta_{i},\mathrm{d}\theta_{i}$ all vanish and $\pi_{i\alpha}$ are linear combinations of $\eta_{1},\eta_{2},\ldots,\eta_{6}$. It follows that an integral manifold of $(\mathcal{P},\mathcal{J})$ can only exist within the locus along which $\tau_{i}$ can be fully absorbed by $\pi_{i\alpha}\wedge\eta_{\alpha}$, as long as one add suitable linear combinations of $\eta_{\alpha}$ to the $1$-forms in $\mathcal{S}$.

This is precisely the approach we take in practice: we keep adjusting the $1$-forms in $\mathcal{S}$ by adding linear combinations of $\eta_{\alpha}$ to them until a point when the rank of $(\tau^{i})$ cannot be further reduced. The remaining $(\tau^{i})$ must vanish along integral manifolds of $(P,\mathcal{J})$.

In our case, the matrix $(\pi_{i\alpha})$ (called the tableau) takes the form

[TABLE]

where $\pi_{1},\ldots,\pi_{16}$ are 1-forms777One can think of $(\pi_{i\alpha})$ as an $\mathcal{A}$-valued $1$-form, where $\mathcal{A}\subset\mathbb{R}^{4}\otimes(\mathbb{R}^{6})^{*}$ is a vector subspace. Here, the $\mathbb{R}$-valued $1$-forms $\pi_{1},\ldots,\pi_{16}$ are just a set of ‘form coordinates’ for $(\pi_{i\alpha})$. linearly independent of $\theta_{1},\ldots,\theta_{4},\eta_{1},\ldots,\eta_{6}$ and among themselves. It is easy to see that all terms in $\tau_{i}$ can be absorbed except the $\eta_{3}\wedge\eta_{4}$ terms in $\tau_{1}$ and $\tau_{2}$ and the $\eta_{1}\wedge\eta_{2}$ terms in $\tau_{3}$ and $\tau_{4}$.

Calculation yields

[TABLE]

[TABLE]

As a result, we have proved:

Lemma 4.1**.**

Integral manifolds of $(\mathcal{P},\mathcal{J})$ must be contained in the locus $\mathcal{L}\subset\mathcal{P}$ defined by the equations

[TABLE]

Definition 4.1**.**

Let $\mathcal{P}_{1}\subset\mathcal{P}$ be the locus defined by the equations

[TABLE]

Let $\mathcal{J}_{1}$ be the pull-back of $\mathcal{J}$ to $\mathcal{P}_{1}$. The rank-4 Pfaffian system $(\mathcal{P}_{1},\mathcal{J}_{1})$ is called the $1$-refined Bäcklund-Pfaff system for rank-$1$ Bäcklund transformations relating two hyperbolic Monge-Ampère systems.

Definition 4.2**.**

A $6$-dimensional integral manifold of the $1$-refined Bäcklund-Pfaff system $(\mathcal{P}_{1},\mathcal{J}_{1})$ is called a $1$-refined lifting of the underlying Bäcklund transformation.

Proposition 4.1**.**

Let $(M,\mathcal{I})$ and $(\bar{M},\bar{\mathcal{I}})$ be as above. Any rank-$1$ Bäcklund transformation $\phi:N\rightarrow M\times\bar{M}$ admits a $1$-refined lifting.

Proof. By the previous discussion, there exists a [math]-refined lifting $\hat{\phi}$ of $\phi$ such that, when pulled-back via $\hat{\phi}$ to $N$,

[TABLE]

If $s_{2}\neq 0$, we transform the pairs $(\alpha_{1},\alpha_{2})^{T}$ and $(\beta_{1},\beta_{2})^{T}$ simultaneously by

[TABLE]

If $s_{2}=0$ but $s_{1}\neq 0$, we transform the pairs $(\alpha_{1},\alpha_{2})^{T}$ and $(\beta_{1},\beta_{2})^{T}$ simultaneously by

[TABLE]

such that the previous case applies. These transformations correspond to choosing new [math]-refined liftings, and the result is a [math]-refined lifting with $s_{1}=0$. In a similar way, we can choose [math]-refined liftings such that, in addition, $t_{3}=0$. This completes the proof. ∎

Now $\mathcal{J}_{1}$ is generated by

[TABLE]

Given a rank-$1$ Bäcklund transformation $\phi:N\rightarrow M\times\bar{M}$, it is easy to see that whether the product $s_{2}t_{4}$ locally vanishes is independent of the choice of $1$-refined liftings of $\phi$. It turns out that the case when $s_{2}t_{4}\equiv 0$ is quite restrictive when both $(M,\mathcal{I})$ and $(\bar{M},\bar{\mathcal{I}})$ are Euler-Lagrange systems.

Proposition 4.2**.**

Let $(M,\mathcal{I})$ and $(\bar{M},\bar{\mathcal{I}})$ be two hyperbolic Euler-Lagrange systems. If there exists a rank-$1$ Bäcklund transformation $\phi:N\rightarrow M\times\bar{M}$ such that $s_{2}t_{4}=0$ on a $1$-refined lifting of $\phi$, then both $(M,\mathcal{I})$ and $(\bar{M},\bar{\mathcal{I}})$ must be contact equivalent to the system representing the wave equation

[TABLE]

Proof. By the Euler-Lagrange assumption and Proposition 2.2, we have $S_{2}=\bar{S}_{2}=\bm{0}$. Let

[TABLE]

By Proposition 2.3, it suffices to show that, on any integral manifold of the $1$-refined Bäcklund-Pfaff system $(\mathcal{P}_{1},\mathcal{J}_{1})$, we have $S_{1}=\bar{S}_{1}=\bm{0}$.

First, we assume that $s_{2}=0$. Restricted to the locus defined by $s_{2}=0$ in $\mathcal{P}_{1}$, the tableau $(\pi_{i\alpha})$ associated to $(\mathcal{P}_{1},\mathcal{J}_{1})$ satisfies

[TABLE]

As a result, for $i,j=3,\ldots,6$ and $i\neq j$, the $\eta^{i}\wedge\eta^{j}$ terms in $\mathrm{d}\theta_{1}$ and $\mathrm{d}\theta_{2}$ cannot be absorbed, and the corresponding coefficients must vanish on any integral manifold of $(\mathcal{P}_{1},\mathcal{J}_{1})$. Calculating with Maple™, we find that

[TABLE]

both congruences being reduced modulo $\theta_{1},\ldots,\theta_{4},\eta_{1},\eta_{2}$. It follows that $S_{1}=\bar{S}_{1}=\bm{0}$.

The case when $t_{4}=0$ is similar. ∎

Proposition 4.3**.**

On any $1$-refined lifting of a rank-$1$ Bäcklund transformation relating two hyperbolic Euler-Lagrange systems, the following expressions must vanish:

[TABLE]

Proof. This is evident when such a Bäcklund transformations satisfies $s_{2}t_{4}=0$. In fact, by Proposition 4.2, the functions $V_{i}$ and $W_{i}$ vanish identically on $\mathcal{P}_{1}$. Otherwise, restricting to the open subset of $\mathcal{P}_{1}$ defined by $s_{2}t_{4}\neq 0$, a calculation similar to that in the proof of Proposition 4.2 shows that the torsion of the Pfaffian system $\mathcal{J}_{1}$ can be absorbed only when $\Phi_{i}$ $(i=1,\ldots,4)$ are all zero. ∎

Corollary 4.2**.**

If two hyperbolic Euler-Lagrange systems are related by a rank-$1$ Bäcklund transformation with $\epsilon=1$, then they are either both degenerate or both nondegenerate.

Proof. We have noted above (see Remark 3) that, if a rank-$1$ Bäcklund transformation relating two hyperbolic Monge-Ampère systems satisfies $\epsilon=1$, then $\mu>1$. The vanishing of $\Phi_{2}$ and $\Phi_{4}$ on a $1$-refined lifting of such a Bäcklund transformation then implies that, on such a lifting,

[TABLE]

By the vanishing of $\Phi_{1}$ and $\Phi_{3}$, it is easy to see that the matrices $S_{1}$ and $\bar{S}_{1}$ are either both degenerate or both nondegenerate. ∎

Note that, in the proof of Corollary 4.2, the condition $V_{2}=W_{2}=0$ is meaningful only if it is independent of the choice of $1$-refined liftings of a Bäcklund transformation. To make this point explicit, we prove the following proposition, which shows how $1$-refined liftings of a rank-$1$ Bäcklund transformation relate to each other.

Proposition 4.4**.**

Let $\phi$ be a rank-$1$ Bäcklund transformation relating two hyperbolic Monge-Ampère systems satisfying $s_{2}t_{4}\neq 0$ on its $1$-refined liftings. There exists a subgroup $H\subset G_{1}\times{G}_{1}$ such that any two $1$-refined liftings of $\phi$ are related, in the $\mathcal{G}_{1}\times\bar{\mathcal{G}}_{1}$ component, by an ${H}$-valued transformation. Moreover,

1. i.

If $\mu>1$, then $H$ is the subgroup generated by elements of the form $h=(g,g^{\prime})$ where

[TABLE]

and $g^{\prime}$ is the result of replacing $h_{4}$ in $g$ by $\epsilon h_{4}$. 2. ii.

If $\mu=1$, then $H$ is generated by the subgroup in case i and the element

[TABLE]

*where $J$ is as in (27) and *$J^{\prime}={\rm diag}(-1,1,-1,1,-1)$J.

Proof. By (40), any two $1$-refined liftings of $\phi$ must be related by a pointwise $H$-action, where $H\subset G_{1}\times G_{1}$ is a subgroup.

Note that $G_{1}$ has two connected components: $G_{e}$ consisting of those elements of the form (26) and $G_{1}\backslash G_{e}=JG_{e}$.

When $\mu>1$, one must have $H\subset G_{e}\times G_{e}$. This is because switching $[\![\alpha_{1},\alpha_{2}]\!]$ with $[\![\alpha_{3},\alpha_{4}]\!]$ enforces switching $[\![\beta_{1},\beta_{2}]\!]$ with $[\![\beta_{3},\beta_{4}]\!]$, which transforms $\mu$ into $\mu^{-1}$, which is not permissible. Now suppose that $h=(g,g^{\prime})\in H$. It is easy to see, by (40) and the assumption $s_{2}t_{4}\neq 0$ that $g\in H$ must be of the form (41). The form of $g^{\prime}$ is then determined, as stated in i.

When $\mu=1$, we have $\epsilon=-1$. In this case, we do allow switching between the indices $(1,2)$ and $(3,4)$, by an action of $J$ on the $\beta$’s and a corresponding action on the $\alpha$’s so that the form of (40) is preserved.

Finally, note that a pointwise action by $H$ described above transforms a $1$-refined lifting of $\phi$ to a $1$-refined lifting of $\phi$.

This completes the proof.∎

Remark 4*.*

By (29) and (30), $h=(g,g^{\prime})$ and $h_{J}=(J^{\prime},J)$ in Proposition 4.4 act on $S_{1}=(V_{i})$ and $\bar{S}_{1}=(W_{i})$ in the following way:

[TABLE]

In particular, if $W_{2}=V_{2}=0$ holds on a $1$-refined lifting of $\phi$, then it holds on any other $1$-refined lifting of $\phi$.

5. A Special Class of Bäcklund Transformations

In the previous section, we have seen that, to a rank-$1$ Bäcklund transformation $\phi:N\rightarrow M\times\bar{M}$ relating two hyperbolic Monge-Ampère systems $(M,\mathcal{I})$ and $(\bar{M},\bar{\mathcal{I}})$, we can associate a function $\mu:N\rightarrow[1,\infty)$ that is independent888In fact, it is easy to see, from the point of view of [Hu19] that $\mu$ is a local invariant of the Bäcklund transformation. of the [math]-refined liftings of $\phi$.

Definition 5.1**.**

A rank-$1$ Bäcklund transformation relating two hyperbolic Monge-Ampère systems is said to be special if $\mu=1$.

For the rest of this section, we will focus on special Bäcklund transformations relating two hyperbolic Euler-Lagrange systems. A motivation for this is that many classical Bäcklund transformations are of this type (cf. [Cle02], [Cle18]).

By Proposition 4.3, given a special rank-$1$ Bäcklund transformation $\phi:N\rightarrow M\times\bar{M}$ relating two hyperbolic Euler-Lagrange systems, the following equalities must hold on any $1$-refined lifting of $\phi$:

[TABLE]

By Remark 4, these conditions are invariant under the $H$-action defined in Proposition 4.4.

Now let $\mathcal{P}_{s}\subset\mathcal{P}_{1}$ be defined by the equations

[TABLE]

Proposition 5.1**.**

Any $1$-refined lifting of a special rank-$1$ Bäcklund transformation relating two hyperbolic Euler-Lagrange systems is completely contained in $\mathcal{P}_{s}$. Moreover, on such a lifting, in addition to (42), we have

[TABLE]

Proof. Restricting to $\mathcal{P}_{s}$, the generators $\theta_{i}$ of $\mathcal{J}_{1}$ satisfy congruences of the form:

[TABLE]

The tableau $(\pi_{i\alpha})$ now takes the form

[TABLE]

where, reduced modulo $\eta_{0},\eta_{1},\ldots,\eta_{4}$,

[TABLE]

By a calculation using Maple™, it is easy to see that the torsion $\tau_{i}$ can be absorbed only if the equations (42) and (43) hold.∎

The equalities (42) and (43) tell us which Euler-Lagrange systems may be special Bäcklund-related. In particular, by Propositions 4.4 and 2.1, it is easy to see that whether $V_{2}$ (hence $W_{2}$) vanishes is independent of the choice of $1$-refined liftings. Thus, we may locally999Namely, the conditions below hold on an entire open subset of $N$. classify special rank-$1$ Bäcklund transformations relating two hyperbolic Euler-Lagrange systems into the following three types:

1. Type I.

$V_{2}=0$, $\det(S_{1})=V_{1}V_{4}\neq 0$; 2. Type II.

$V_{2}=0$, $\det(S_{1})=V_{1}V_{4}=0$; 3. Type III.

$V_{2}\neq 0$.

Proposition 5.2**.**

$\bf(A)$* If a pair of hyperbolic Euler-Lagrange systems are related by a Type I special Bäcklund transformation, then one of them must be positive, the other negative.

$\bf(B)$ Two hyperbolic Euler-Lagrange systems related by a Type III special Bäcklund transformation cannot be both degenerate.*

Proof. For Part $\bf(A)$, it is immediate by (42) that, on a $1$-refined lifting, $\det(S_{1})=V_{1}V_{4}=-W_{1}W_{4}=-\det(\bar{S}_{1})\neq 0$. Therefore, one of the Euler-Lagrange systems being Bäcklund-related is positive, the other negative.

To prove Part $\bf(B)$, first apply Proposition 4.4 to show that, in this case, one can always find a $1$-refined lifting on which $V_{1}=V_{4}=0$. (By Remark 4, this can be achieved by acting on an initial $1$-refined lifting by an element $h\in H$ with $h_{0}=h_{1}=h_{2}=1$, $h_{3}=V_{4}/V_{2}$, $h_{4}=-V_{1}/V_{2}$.) For such a $1$-refined lifting, by (43), the two Euler-Lagrange systems can be both degenerate only when $s_{2}t_{4}=0$. By Proposition 4.2, both $S_{1}$ and $\bar{S}_{1}$ must vanish, which impossible since we have assumed $V_{2}\neq 0$. ∎

Now we focus on Type $\rm II$.

Let $\phi:N\rightarrow M\times\bar{M}$ be a Type II special rank-$1$ Bäcklund transformation in the sense above. By Propositions 2.1 and 4.4, whether the pair $(V_{1},V_{4})$ vanishes is independent of the choice of $1$-refined liftings of $\phi$. It follows that $\phi$ must be one of the following two types.

Type IIa: on any $1$-refined lifting of $\phi$, $(V_{1},V_{4})=0$;

Type IIb: on any $1$-refined lifting of $\phi$, $(V_{1},V_{4})\neq 0$.

5.1. Type IIa

In this case, (42) implies that

[TABLE]

If locally either $V_{3}$ or $W_{3}$ is zero, which is independent of the choice of $1$-refined liftings, the underlying Bäcklund transformation must relate a hyperbolic Euler-Lagrange system with the system corresponding to $z_{xy}=0$. See [CI09] and [Zvy91] for a classification of all hyperbolic Monge-Ampère systems that are rank-$1$ Bäcklund-related to the equation $z_{xy}=0$.

More generally, we have the following theorem.

Theorem 5.1**.**

If two hyperbolic Euler-Lagrange systems are related by a Type IIa special rank-$1$ Bäcklund transformation, then each of them corresponds (up to contact equivalence) to a second order PDE of the form

[TABLE]

Proof. By definition, any Type IIa special Bäcklund transformation admits a $1$-refined lifting that is completely contained in the locus $\mathcal{P}_{{\rm IIa}}\subset\mathcal{P}_{1}$ defined by the equations

[TABLE]

Let $\mathcal{G}_{2}\subset\mathcal{G}_{1}$ be the subbundle defined by $V_{1}=V_{2}=V_{4}=0$; similarly, let $\bar{\mathcal{G}}_{2}\subset\bar{\mathcal{G}}_{1}$ be the subbundle defined by $W_{1}=W_{2}=W_{4}=0$. It is clear that $\mathcal{P}_{{\rm IIa}}$ is the product of $\mathcal{G}_{2}$, $\bar{\mathcal{G}}_{2}$, and a space of parameters with coordinates $(s_{2},t_{4})$.

The calculations below are performed using Maple™.

By (33), on $\mathcal{G}_{2}$, there exist functions $P_{ij}$ and $V_{3j}$ such that

[TABLE]

Similarly, on $\bar{\mathcal{G}}_{2}$, there exist functions $Q_{ij}$ and $W_{3j}$ such that

[TABLE]

There is freedom to add linear combinations of $\alpha_{0},\ldots,\alpha_{4}$ (resp. $\beta_{0},\ldots,\beta_{4}$) into $\varphi_{i}$ (resp. $\psi_{i}$) without changing the form of the corresponding Monge-Ampère structure equations. Using this, we can arrange the following expressions to be zero:

[TABLE]

Applying $\mathrm{d}^{2}=0$ to the Monge-Ampère structure equations yields that

[TABLE]

This implies that, on $\mathcal{G}_{2}$,

[TABLE]

By a similar argument, one can show that, on $\bar{\mathcal{G}}_{2}$,

[TABLE]

Restricted to $\mathcal{P}_{{\rm IIa}}$, the generators $\theta_{i}$ $(i=1,\ldots,4)$ of $\mathcal{J}_{1}$ satisfy congruences of the form

[TABLE]

The tableau takes the form:

[TABLE]

where, modulo $\eta_{i}$,

[TABLE]

Assuming $s_{2},t_{4}$ to be both nonzero, we compute and find that the torsion can be absorbed only if the following expressions are zero:

[TABLE]

One can verify that, on the subbundle of $\mathcal{G}_{2}$ defined by the vanishing of $P_{20},P_{23},P_{24},$ $P_{60},P_{61}$ and $P_{62}$, the following structure equations hold:

[TABLE]

Clearly, the systems $\langle\alpha_{1}\rangle$ and $\langle\alpha_{3}\rangle$ are both integrable. It is a similar case for the structure equations on $\bar{\mathcal{G}_{2}}$. By Proposition 2.4, the proof is complete. ∎

5.2. Type IIb

In this case, on a $1$-refined lifting of $\phi$, either $V_{1}$ or $V_{4}$ vanishes. By Proposition 4.4 (in particular, using $h_{J}$), we can arrange $V_{1}\neq 0$ and $V_{4}=0$ on a $1$-refined lifting of $\phi$. Such a $1$-refined lifting can always be chosen to further satisfy $V_{1}=1$ and $V_{3}=W_{3}=0$ or $1$.

In the next proposition we show that the case of $V_{1}=1$ and $V_{3}=W_{3}=0$ is impossible. Then we characterize the case when $\phi$ admits a $1$-refined lifting for which $V_{1}=V_{3}=W_{3}=1$.

Proposition 5.3**.**

Restricting to the locus in $\mathcal{P}_{1}$ defined by

[TABLE]

$\mathcal{J}_{1}$* has no integral manifold.*

Proof. If there exists a $1$-refined lifting of a special Bäcklund transformation such that $V_{3}=W_{3}=0$, then the equality (43) enforces that $s_{2}t_{4}=0$ on such a lifting. By Proposition 4.2, both Monge-Ampère systems must be contact equivalent to the wave equation $z_{xy}=0$. In particular, $V_{i}$ and $W_{i}$ must all be zero on $\mathcal{G}_{1}$ and $\bar{\mathcal{G}_{1}}$, respectively. This contradicts our assumption. ∎

Theorem 5.2**.**

Let $(M,\mathcal{I})$ and $(\bar{M},\bar{\mathcal{I}})$ be two hyperbolic Euler-Lagrange systems. If $\phi:N\rightarrow M\times\bar{M}$ defines a Type IIb special rank-$1$ Bäcklund transformation relating $(M,\mathcal{I})$ and $(\bar{M},\bar{\mathcal{I}})$, then each of $(M,\mathcal{I})$ and $(\bar{M},\bar{\mathcal{I}})$ must have a characteristic system that contains a rank-1 integrable subsystem.

Proof. The idea is similar to that of Theorem 5.1. We restrict the differential ideal $\mathcal{J}_{1}$ to the locus $\mathcal{P}_{{\rm IIb}}\subset\mathcal{P}_{1}$ defined by the equations

[TABLE]

and analyze the obstructions to integrability of the resulting rank-$4$ Pfaffian system.

The calculations below are performed using Maple™.

By (33), on the subbundle of $\mathcal{G}_{1}$ defined by $V_{1}=V_{3}=1$ and $V_{2}=V_{4}=0$, there exist functions $P_{ij}$ such that

[TABLE]

Using the freedom in the choice of $\varphi_{i}$, we can arrange that

[TABLE]

are zero.

Expanding $\mathrm{d}(\mathrm{d}\alpha_{i})=0$, we find that

[TABLE]

Similarly, on the subbundle of $\bar{\mathcal{G}}_{1}$ defined by $-W_{1}=W_{3}=1$ and $W_{2}=W_{4}=0$, there exist functions $Q_{ij}$ such that

[TABLE]

Using the freedom in the choice of $\psi_{i}$, we can arrange that

[TABLE]

are zero.

By expanding $\mathrm{d}(\mathrm{d}\beta_{i})=0$, we find that

[TABLE]

Denote the restriction of ${\mathcal{J}}_{1}$ to $\mathcal{P}_{{\rm IIb}}$ by ${\mathcal{J}}_{\rm IIb}$. By computation, we find that the torsion of $(\mathcal{P}_{{\rm IIb}},\mathcal{J}_{\rm IIb})$ can be absorbed only if the following expressions are zero:

[TABLE]

In particular, the vanishing of $P_{60},P_{62},Q_{60}$ and $Q_{62}$ implies that

[TABLE]

The conclusion of the proposition follows. ∎

6. New Examples of Type III

In this section, we present some new examples of Type III special rank-$1$ Bäcklund transformations. Their existence shows that a degenerate hyperbolic Euler-Lagrange system may be special Bäcklund-related to a non-degenerate one. One of these examples (Section 6.2) is, up to contact transformations, a Bäcklund transformation relating solutions of the PDE

[TABLE]

to those of a more complicated PDE of the form (3) whose coefficients are given by (65) in Appendix B. We note that (45) is on the Goursat-Vessiot list101010See [Gou99], [Ves39] and [Ves42]. The recent [CI09] has a summary of the list. of PDEs of the form

[TABLE]

that are Darboux-integrable at the 2-jet level.

6.1. A Class of New Examples

Using a method in [Hu19] and after a somewhat lengthy calculation using Maple™, we obtain on a 6-manifold $N$ involutive111111Namely, exterior differentiation applied to these equations yields identities. structure equations (see Appendix A for details) of the form:

[TABLE]

[TABLE]

where $C^{i}_{jk}=-C^{i}_{kj},R_{k},S_{k},T_{k}$ are certain fixed analytic functions defined on $\Omega:=\{RST\neq 0\}\subset\mathbb{R}^{3}$.

One can verify that each of the ideals

[TABLE]

is generated by the pull-back of a hyperbolic Monge-Ampère ideal ($\mathcal{I}_{0}$ and $\bar{\mathcal{I}}_{0}$, resp.) defined on a 5-dimensional quotient ($M$ and $\bar{M}$, resp.) of $N$.

By

[TABLE]

it is immediate that $N$ is a special rank-$1$ Bäcklund transformation relating $(M,\mathcal{I}_{0})$ and $(\bar{M},\bar{\mathcal{I}}_{0})$. Furthermore, we choose the new bases

[TABLE]

and

[TABLE]

such that their pull-backs via arbitrary sections $\iota:M\rightarrow N$ and $\bar{\iota}:\bar{M}\rightarrow N$, respectively, are $1$-adapted hyperbolic Monge-Ampère coframings.

Computing using these new bases, we find that

[TABLE]

[TABLE]

Clearly, this tells us that both $\mathcal{I}_{0}$ and $\bar{\mathcal{I}}_{0}$ are hyperbolic Euler-Lagrange (Proposition 2.2). Moreover, since $\det(S_{1})=0$ and $\det(\bar{S_{1}})=2T\neq 0$, $\mathcal{I}_{0}$ is degenerate, and $\bar{\mathcal{I}}_{0}$ is non-degenerate, by our definition.

By Cartan’s third theorem (see [Bry14]), for each $u\in\mathbb{R}^{6}$ and for any $(R_{0},S_{0},T_{0})\in\Omega$, there exists a coframing $(\omega^{i})$ and functions $R,S,T$ on a neighborhood $U$ of $u$ such that the structure equations (46) and (47) hold with $(R,S,T)(u)=(R_{0},S_{0},T_{0})$.

This fact tells us that $T$ may be positive or negative. In other words, $\bar{\mathcal{I}}_{0}$ can be a hyperbolic Euler-Lagrange system of either the positive or the negative type.

Note that, generically, the map $(R,S,T):N^{6}\rightarrow\mathbb{R}^{3}$ has rank $3$. This is easy to see by computing the differentials $\mathrm{d}(R),\mathrm{d}(S)$ and $\mathrm{d}(T)$. In this generic case, $N$ represents a Bäcklund transformation of cohomogeneity $3$.

On the other hand, the map $(R,S,T)$ has rank $2$, which is the minimum rank possible, if and only if

[TABLE]

By the structure equations, $\mathrm{d}(T-2R^{2}S^{2})$ vanishes whenever $T-2R^{2}S^{2}$ does.

This gives rise to the condition when the underlying Bäcklund transformation has cohomogeneity $2$, that is, when $T=2R^{2}S^{2}$. In particular, since $T>0$ in this case, $\bar{\mathcal{I}}_{0}$ must be of the positive type. We present further analysis of this case below.

6.2. The $T=2R^{2}S^{2}$ Case

All calculations below, unless otherwise noted, are performed using Maple™.

Setting $T=2R^{2}S^{2}$, the structure equations (46) and (47) now take the form

[TABLE]

[TABLE]

where $\tilde{C}^{i}_{jk}(R,S)=C^{i}_{jk}(R,S,2R^{2}S^{2})$, and similarly for $\tilde{R}_{k}$ and $\tilde{S}_{k}$.

Let $\sigma^{i}$ and $\tau^{i}$ be as in (48) and (49), respectively. We find that the $1$-form

[TABLE]

is exact. Thus, locally there exists a function $f$ on $N$ such that

[TABLE]

Now consider the $1$-forms

[TABLE]

Using (50) and (51) and by letting $U=e^{f}/S$, we obtain the following structure equations for $(\xi^{i})$:

[TABLE]

It is clear from these structure equations that $\xi^{i}$ $(i=0,\ldots,4)$ are well-defined on a $5$-dimensional quotient $M^{5}$ of $N$, the corresponding hyperbolic Euler-Lagrange system being $(M,\mathcal{I}_{0})$, where

[TABLE]

In particular, note that $\xi^{1}$ and $\xi^{3}$ are both integrable. By Proposition 2.4, $(M,\mathcal{I}_{0})$ is contact equivalent to a PDE of the form $z_{xy}=F(x,y,z,z_{x},z_{y})$.

In Appendix B, we find local coordinates for $(M,\mathcal{I}_{0})$ and prove that it is contact equivalent to the equation:

[TABLE]

On the other hand, consider the $1$-forms

[TABLE]

Using (50) and (51), we find that

[TABLE]

It is clear, by these structure equations, that $(\xi^{i})$ descend to a coframing on a $5$-dimensional quotient $\bar{M}^{5}$ of $N$, the corresponding hyperbolic Euler-Lagrange system being $(\bar{M}^{5},\bar{\mathcal{I}}_{0})$, where

[TABLE]

We have found local coordinates on $\bar{M}$ that put $(\bar{M},\bar{\mathcal{I}}_{0})$ (up to contact transformations) in a PDE form. Since both the expression and its derivation are rather complicated, we include them in Appendix B.

Remark 5*.*

This example shows the possibility for a pair of hyperbolic Euler-Lagrange systems of distinct types to be special Bäcklund-related, in which one admits nontrivial first integrals for its characteristic systems, while the other doesn’t. In contrast, suppose that $S$ and $\bar{S}$ are, respectively, integral surfaces of two hyperbolic Monge-Ampère systems $(M,\mathcal{I})$ and $(\bar{M},\bar{\mathcal{I}})$. If $S,\bar{S}$ are related to each other by a rank-$1$ Bäcklund transformation, then the characteristic curves in $S$ correspond, under the Bäcklund transformation, to the characteristic curves in $\bar{S}$, and vice versa. This is easy to see by the condition $(2)$ in Definition 1.3.

7. Some Classical Examples

In this section, we present some examples of special rank-$1$ Bäcklund transformations. These examples are not new. We hope they can serve as a motivation for the open questions described in the next section.

1. 1.

A class of homogeneous rank-$1$ Bäcklund transformations. In [Cle02], it is shown that, if, on $N^{6}$, there exists a local coframing $(\theta_{1},\theta_{2},\omega^{1},\ldots,\omega^{4})$ satisfying

[TABLE]

where $B_{i}$ are constants and $\sigma=\pm 1$, then the systems

[TABLE]

are hyperbolic Monge-Ampère systems well-defined on some $5$-dimensional quotients of $N$. Here $N$ is a homogeneous rank-$1$ Bäcklund transformation relating the two Monge-Ampère systems.

It is clear, by the equation (see Lemma 3.1)

[TABLE]

that $\epsilon\mu^{4}=-1$. Therefore, these Bäcklund transformations are special.

In particular, when $\sigma=-1$, $B_{1}=-(\tau+1)^{2}/2$ and $B_{3}=(\tau^{-1}+1)^{2}/2$ for some constant $\tau\neq 0$, one obtains the classical Bäcklund transformation between $K=-1$ surfaces in $\mathbb{E}^{3}$ when $\tau>0$ and a Bäcklund transformation between $K=1$ surfaces in $\mathbb{E}^{2,1}$ when $\tau<0$. 2. 2.

Let $(x,y,u,p,q)$ and $(X,Y,v,P,Q)$ be coordinates on the spaces of two sine-Gordon systems:

[TABLE]

The locus $N$ in the product manifold defined by the equations

[TABLE]

corresponds to the Bäcklund transformation (2).

The pull-backs to $N$ of $\theta=\mathrm{d}u-p\mathrm{d}x-q\mathrm{d}y$ and $\bar{\theta}=\mathrm{d}v-P\mathrm{d}X-Q\mathrm{d}Y$ satisfy

[TABLE]

It follows that the Bäcklund transformation (2) is special.

8. Open Questions

Several results concerning special Bäcklund transformations relating two hyperbolic Euler-Lagrange systems can be summarized in the diagram (Figure 1) below.

1. a.

An immediate question is: Can a special rank-$1$ Bäcklund transformation belong to region $6$ in the diagram? 2. b.

Does there exist any special rank-$1$ Bäcklund transformation in regions $1$ and $3$ that does not relate an Euler-Lagrange system to itself? 3. c.

A more subtle problem is to understand the generality of existence in each of the regions. For instance, it would be interesting to determine integers $k,\ell$ such that: The space of special rank-$1$ Bäcklund transformations relating two positive hyperbolic Euler-Lagrange systems is parametrized by $k$ functions of $\ell$ variables. 4. d.

What can be said about the existence of non-special rank-$1$ Bäcklund transformations relating two hyperbolic Euler-Lagrange systems? Is there any hyperbolic Euler-Lagrange pair that are rank-$1$ Bäcklund-related but not special rank-$1$ Bäcklund-related? 5. e.

When can a hyperbolic Euler-Lagrange system be Bäcklund-related to a non-Euler-Lagrange hyperbolic Monge-Ampère system?

9. Acknowledgement

The author would like to thank his thesis advisor, Prof. Robert Bryant, for all his support and guidance in research and Prof. Jeanne N. Clelland for her advice during the preparation of the current work for publication. He would like to thank the referee for giving many constructive comments, which motivated the writing of Appendix C.

Appendix A Table of Structure Functions

In this appendix, we record the functions $C^{i}_{jk}=-C^{i}_{kj},R_{k},S_{k},T_{k}$ involved in the equations (46) and (47).

Appendix B $(M,\mathcal{I}_{0})$ and $(\bar{M},\bar{\mathcal{I}}_{0})$ in PDE Forms

In this Appendix, we integrate the structure equations (52)-(58) for $(M,\mathcal{I}_{0})$ and (59)-(64) for $(\bar{M},\bar{\mathcal{I}}_{0})$. By ‘integrate’, we mean finding local coordinates, expressing the coframings and structure functions in terms of these coordinates such that the structure equations hold identically. As a result, we put these hyperbolic Euler-Lagrange systems in PDE forms (up to contact transformations). Most calculations below are performed using Maple™.

I. Integration of $(M,\mathcal{I}_{0})$.

Consider (52)-(58). We integrate these structure equations in the following steps.

First note that these structure equations are invariant under the transformation:

[TABLE]

As a result, we can assume that $R>0$. 2. 2)

It is clear that

[TABLE]

Thus, both $\xi^{1}$ and $\xi^{3}$ are multiples of exact forms. In fact, we find that the $1$-forms

[TABLE]

satisfy

[TABLE]

and

[TABLE]

It follows that there exist local coordinates $x,y,r,s$ such that

[TABLE]

with freedom to add constants to $r,s,x,y$, simultaneously scaling $x,y$. 3. 3)

It is straightforward to verify that

[TABLE]

Using the freedom of adding a constant to $r$ or $s$, we can arrange that

[TABLE]

moreover, there exists a constant $\lambda\neq 0$ such that $e^{r}R=\lambda U$. (At this point, we choose not to normalize $\lambda$.) Using these and the expression of $\mathrm{d}U$, we obtain

[TABLE]

Direct integration gives

[TABLE]

for some constant $C$. We can normalize $C$ to be zero by adding a constant to $x$ or $y$. 4. 4)

Since

[TABLE]

the system $\langle\xi^{0},\xi^{2},\xi^{4}\rangle$ is Frobenius. We proceed by looking for its first integrals.

Indeed, we find that

[TABLE]

satisfies

[TABLE]

Hence, $\Phi$ is a multiple of an exact form. By adding an appropriate multiple of $\Phi$ into $\frac{R^{2}-1}{R}(\xi^{1}+\xi^{3})$, we find that

[TABLE]

Therefore, locally there exists a function $v$ such that

[TABLE] 5. 5)

One can verify that the functions $R,x,y,v$ do not have linearly independent differentials. In fact, we find that

[TABLE]

Setting $G(x,y,z):=(R^{2}+1)^{-1}$, direct integration gives

[TABLE]

where $C_{1}$ is a constant. We can normalize $C_{1}$ to be zero by adding constants to $x$ and $y$. For convenience, we express $v$ in terms of $R,x,y$. 6. 6)

Next, we observe that the $1$-forms

[TABLE]

satisfy

[TABLE]

Consequently, there exist functions $g,h$ such that

[TABLE] 7. 7)

At this point, each $\xi^{i}$ as well as the function $U$ can be expressed in terms of $x,y,R,g,h$ and their exterior derivatives. Moreover, (52)-(58) become identities.

In particular, one can already put $\xi^{0}$ in the form

[TABLE]

by introducing certain functions $z,p,q$. In our choice,

[TABLE]

Computing using these coordinates, we obtain

[TABLE]

This tells us that $(M,\mathcal{I}_{0})$ is contact equivalent to the PDE

[TABLE]

which is, after we scale $x$ and $y$, contact equivalent to

[TABLE]

II. Integration of $(\bar{M},\bar{\mathcal{I}}_{0})$.

We integrate the structure equations (59)-(64) in the following steps.

First note that

[TABLE]

Thus, the system $\langle\eta^{2},\eta^{4}\rangle$ is Frobenius. In particular, there exists a linear combination of $\eta^{2},\eta^{4}$ that is a multiple of an exact $1$-form. Setting undetermined weights that are functions of $R$, we find that the $1$-form

[TABLE]

satisfies

[TABLE]

for some $1$-form $\psi$.

By adding an appropriate multiple of $\Phi$ to $\psi$, we obtain an exact $1$-form (still denoted by $\psi$):

[TABLE]

Thus, locally there exist functions $f,g$ such that

[TABLE] 2. 2)

Next, we compute and observe that

[TABLE]

This motivates the calculation:

[TABLE]

where

[TABLE]

We constructed this $\gamma$, by choosing an appropriate ‘$\mathrm{d}g$’-term, so that it is exact.

Consequently, there exist functions $u,v$ such that

[TABLE] 3. 3)

Now each $\eta^{i}$ can be expressed completely in terms of $f,g,R,u,v$ and their differentials. Moreover, (59)-(64) become identities.

In particular,

[TABLE]

By the Pfaff theorem, up to scaling, we can put $\eta^{0}$ in the form

[TABLE]

We approach this by following the proof of the Pfaff theorem in Chapter II of [BCG*+*13]. We first expand $\mathrm{d}\eta^{0}\wedge\eta^{0}\wedge\mathrm{d}R$ in coordinates; from its expression we find that

[TABLE]

satisfies

[TABLE]

Next, writing $u$ in terms of $x,R$ and $g$, the $3$-form $\eta^{0}\wedge\mathrm{d}R\wedge\mathrm{d}x$ has only $3$ terms in it. This allows us to find

[TABLE]

which satisfies

[TABLE] 4. 4)

It follows from the previous step that $\eta^{0}$ is a linear combination of $\mathrm{d}x,\mathrm{d}y$ and $\mathrm{d}R$. After scaling $\eta^{0}$ such that the coefficient of $\mathrm{d}R$ is $1$, we obtain the $1$-form

[TABLE]

where

[TABLE]

and

[TABLE] 5. 5)

It is now reasonable to set $z=R$. Using these coordinates, we compute and observe that

[TABLE]

where

[TABLE]

The PDE form of $(\bar{M},\bar{\mathcal{I}}_{0})$ (up to contact transformations) is therefore (3) in the introduction, where $A,B,C,D,E$ are the same as the above with $p,q$ being replaced by $z_{x}$ and $z_{y}$, respectively.

Furthermore, hyperbolicity demands that

[TABLE]

in other words, the domain of the variables $(x,y,z,p,q)$ needs to satisfy $q\neq 0$ and $2p^{2}z^{2}+2p^{2}+zq>0$.

Appendix C A Note on Calculation

This appendix is written for those readers who wish to see certain concepts and examples in coordinates. In item I, we provide a program that computes the hyperbolic Monge-Ampère relative invariants $S_{1}$ and $S_{2}$ in coordinates. In items II, III, we focus on hyperbolic Euler-Lagrange systems and their types.

I. The Monge-Ampère Relative Invariants (28).

The Monge-Ampère equation (3) corresponds to the exterior differential system $(J^{1}(\mathbb{R}^{2},\mathbb{R}),\langle\theta,\mathrm{d}\theta,\Omega\rangle_{\rm alg})$ with

[TABLE]

If $A,B,D,E$ are all zero, then the system is either empty or equivalent to the wave equation $z_{xy}=0$. From now on, we assume that not all of $A,B,D,E$ are zero. Under this assumption, we could always make one of the following contact transformations to arrange that $E\neq 0$:

$(x^{\prime},y^{\prime},z^{\prime},p^{\prime},q^{\prime})=(p,q,z-px-qy,-x,-y)$; 2. 2)

$(x^{\prime},y^{\prime},z^{\prime},p^{\prime},q^{\prime})=(p,y,z-px,-x,q)$; 3. 3)

$(x^{\prime},y^{\prime},z^{\prime},p^{\prime},q^{\prime})=(x,q,z-qy,x,-y)$.

It then follows that, by scaling the equation, we can arrange that $E=1$. In doing this, the hyperbolicity assumption would allow us to express

[TABLE]

for some $\mu>0$.

By introducing the $1$-forms:

[TABLE]

we obtain:

[TABLE]

In other words,

[TABLE]

Letting $\eta^{0}=2\mu\theta$, we have

[TABLE]

However, the coframing $(\eta^{0},\eta^{1},\ldots,\eta^{4})$ is not necessarily $1$-adapted (see Section 2).

Now define the coefficients $T^{i}_{jk}=-T^{i}_{kj}$ by

[TABLE]

In particular, let121212We remark that each $c^{i}$ has 21 terms in $B,C,D,\mu$, their partial derivatives, and $p,q$.

[TABLE]

Construct a new coframing $(\omega^{0},\omega^{1},\ldots,\omega^{4})$ by

[TABLE]

One can verify that $(\omega^{0},\omega^{1},\ldots,\omega^{4})$ is $1$-adapted; thus, it can be used to compute the Monge-Ampère relative invariants $V_{1},\ldots,V_{8}$.

In fact, defining $\tilde{T}^{i}_{jk}=-\tilde{T}^{i}_{kj}$ using

[TABLE]

we obtain

[TABLE]

[TABLE]

Such expressions of $V_{i}$ depend on up to the second partial derivatives of $B,C,D,\mu$ and are rather complicated; to give the reader a sense, in our calculation, $V_{1},\ldots,V_{8}$ have 476, 159, 159, 476, 155, 262, 155 and 262 terms in them, respectively. We have used the following code in our calculation (with Maple™and the Cartan package), where M stands for the function $\mu$ above.

restart:

with(Cartan):

unprotect(D):

Omega:= Ad(p)&$\wedge$d(q) + Bd(p)&$\wedge$d(y) + C*(d(x)&$\wedge$d(p)- d(y)&$\wedge$d(q))

• Dd(x)&$\wedge$d(q) + Ed(x)&$\wedge$d(y):

Theta:= d(x)&$\wedge$d(p) + d(y)&$\wedge$d(q):

E:=1:

A:= B*D - C$\wedge$2 + M$\wedge$2:

var[1]:= x: var[2]:= y: var[3]:= z: var[4]:= p: var[5]:= q:

for t in [B,C,D,M] do:

d(t):= add(t[i]*d(var[i]), i = 1..5):

od:

for t in [B,C,D,M] do:

for i from 1 to 5 do:

d(t[i]):= add(t[i,j]*d(var[j]), j = 1..5):

od:

od:

for t in [B,C,D,M] do:

d(d(t)):Simf(%):ScalarForm(%):solve(%):assign(%):

od:

for i from 0 to 4 do:

Form(eta[i] = 1):

od:

eta2coord:= {eta[0] = (d(z)- pd(x) - qd(y))(2M),

eta[1] = (M + C)d(p) + Dd(q) + d(y),

eta[2] = -B*d(p) + (M - C)*d(q) - d(x),

eta[3] = (-M + C)d(p) + Dd(q) + d(y),

eta[4] = B*d(p) + (M + C)*d(q) + d(x)}:

coord2eta:= solve(eta2coord, {d(z), d(x), d(y), d(p), d(q)}):

for i from 0 to 4 do:

d(subs(eta2coord, eta[i])):Simf(%):subs(coord2eta,%):deta[i]:=Simf(%):

for j from 0 to 3 do:

for k from j+1 to 4 do:

T[i,j,k]:= pick(deta[i], eta[j], eta[k]):

od:

od:

od:

c1:= T[1,3,4]:

c2:= T[2,3,4]:

c3:= T[3,1,2]:

c4:= T[4,1,2]:

for i from 0 to 4 do:

Form(omega[i] = 1):

od:

assign(eta2coord):

om2coord:= {omega[0] = eta[0],

omega[1] = eta[1] - c1*eta[0],

omega[2] = eta[2] - c2*eta[0],

omega[3] = eta[3] - c3*eta[0],

omega[4] = eta[4] - c4*eta[0]}:

coord2om:= solve(om2coord, {d(z), d(x), d(y), d(p), d(q)}):

for i from 0 to 4 do:

d(subs(om2coord, omega[i])):Simf(%):subs(coord2om,%): dom[i]:= Simf(%):

for j from 0 to 3 do:

for k from j+1 to 4 do:

T[i,j,k]:= pick(dom[i], omega[j], omega[k]):

od:

od:

od:

V[1]:= (T[1,0,3] - T[4,0,2])/2: V[2]:= (T[1,0,4] + T[3,0,2])/2:

V[3]:= (T[2,0,3] + T[4,0,1])/2: V[4]:= (T[2,0,4] - T[3,0,1])/2:

V[5]:= (T[1,0,3] + T[4,0,2])/2: V[6]:= (T[1,0,4] - T[3,0,2])/2:

V[7]:= (T[2,0,3] - T[4,0,1])/2: V[8]:= (T[2,0,4] + T[3,0,1])/2:

We remind the reader that $S_{1},S_{2}$ are relative invariants under contact transformations; they transform by (29) and (30).

II. Euler-Lagrange Examples Revisited.

Given a hyperbolic Monge-Ampère PDE (3), the code above can be used to decide whether it is Euler-Lagrange and, when it is, its Euler-Lagrange type (positive, negative, degenerate).

For the examples 1 and 3 in Section 2, we remind the reader of their classical PDE forms; then, we continue the calculation above by computing $S_{2}$ and $\det(S_{1})$ for these examples as well as for the Monge-Ampère equation satisfying (65).

1. 1.

$K=-1$** Surfaces in $\mathbb{E}^{3}$.** (Example 1, Section 2)

Suppose that a surface $S\subset\mathbb{E}^{3}$ is a graph with the position vector ${\bf x}=(x,y,z(x,y))$. Letting $p:=z_{x}$ and $q:=z_{y}$, we have that, pulled back to $S$,

[TABLE]

The oriented unit normal along $S$ is just

[TABLE]

Direct calculation yields the area form

[TABLE]

and

[TABLE]

where $K$ is the Gauss curvature of $S$.

It follows that the system for $K=-1$ is characterized by

[TABLE]

Noting that $\mathrm{d}x\wedge\mathrm{d}y\neq 0$ on $S$, the PDE form of this system is:

[TABLE] 2. 2.

$K=1$** Time-like Surfaces in $\mathbb{E}^{2,1}$.** (Example 3, Section 2)

Since we use the signature $(+,-,+)$ for the Lorentzian metric on $\mathbb{E}^{2,1}$, the dot and cross products are defined, respectively, by

[TABLE]

Let $S$ be a time-like surface with the position vector ${\bf x}$ and a pseudo-orthonormal frame field $({\bf e}_{1},{\bf e}_{2},{\bf e}_{3})$ attached to it (${\bf e}_{1},{\bf e}_{2}$ being tangent to $S$). The area form $\mathrm{d}\mathcal{A}$ and the Gauss curvature $K$ can be computed by

[TABLE]

and

[TABLE]

where the $2$-forms on the right-hand-sides are their pull-backs to $S$.

When $S$ is a graph ${\bf x}=(x,y,z(x,y))$, it is easy to find that

[TABLE]

Direct calculation yields that

[TABLE]

Therefore, the equation for $K=1$ is

[TABLE]

Now, calculating using Maple™, we find:131313Note that, in (65), $2p^{2}z^{2}+2p^{2}+zq>0$, by hyperbolicity.

[TABLE]

The Euler-Lagrange types are consistent with our earlier observation when calculation was made using differential forms. Here, we have used the following code for computing $\det(S_{1})$ and $S_{2}$ associated to (65), which can be easily modified to work for (66) and (67).

ABCDE:= {eA = 2qz*(z$\wedge$2+1)$\wedge$3,

eB = 2q$\wedge$2(z$\wedge$2+1)$\wedge$2*(4p$\wedge$2z$\wedge$3 - qz$\wedge$2 + 4p$\wedge$2z + 3q),

eC = -2pq*(z$\wedge$2+1)$\wedge$3*(4p$\wedge$2z+q),

eD = (z$\wedge$2+1)(4p$\wedge$2z$\wedge$3+qz$\wedge$2+4p$\wedge$2z - q)(2p$\wedge$2z$\wedge$2+qz+2*p$\wedge$2),

eE = -q$\wedge$3*(4p$\wedge$2z$\wedge$5+qz$\wedge$4 - 16p$\wedge$2z$\wedge$3 - 8qz$\wedge$2 - 20p$\wedge$2*z - q)}:

BCDM:= subs(ABCDE, {B = eB/eE, C = eC/eE, D = eD/eE,

M = sqrt((eAeE + eC$\wedge$2 - eBeD))/eE}):

dBCDM:= {}: ddBCDM:= {}:

for t in [B,C,D,M] do:

for i from 1 to 5 do:

dBCDM:= {op(dBCDM), t[i] = diff(subs(BCDM, t), var[i])}:

od:

od:

for t in [B,C,D,M] do:

for i from 1 to 5 do:

for j from 1 to 5 do:

ddBCDM:= {op(ddBCDM), t[i,j] = diff(diff(subs(BCDM, t), var[i]), var[j])}:

od:

od:

od:

for i from 5 to 8 do:

simplify(subs(BCDM union dBCDM union ddBCDM, V[i]));

od;

subs(BCDM union dBCDM union ddBCDM, V[1]*V[4]-V[2]*V[3]):simplify(%)

III. The Lagrangian.

Classically, a central object in a 2D variational problem is the functional:

[TABLE]

where $L$ is the Lagrangian (function) and $\mathcal{D}\subset\mathbb{R}^{2}$ a fixed compact domain.

The Euler-Lagrange equation associated to a fixed-boundary variation is

[TABLE]

A coordinate-independent formulation (see [BGG03, Section 1.2]) considers instead a fixed boundary variation in $J^{1}(\mathbb{R}^{2},\mathbb{R})$ (with standard coordinates $(x,y,z,p,q)$) by Legendre surfaces141414That is, surfaces that annihilate the pull-back of $\mathrm{d}z-p\mathrm{d}x-q\mathrm{d}y$. $S$ and the functional:

[TABLE]

where $\Lambda=L(x,y,z,p,q)\mathrm{d}x\wedge\mathrm{d}y$. This variational problem may seem less restrictive than the classical one,151515Note that a fixed-boundary variation in the classical formulation may not lift to be a fixed-boundary variation by Legendre surfaces in $J^{1}(\mathbb{R}^{2},\mathbb{R})$. but the corresponding Euler-Lagrange equation is the same as (68), which is Monge-Ampère (3) with

[TABLE]

[TABLE]

More generally but still in the 2D case, $\Lambda$ can be any $2$-form on $J^{1}(\mathbb{R}^{2},\mathbb{R})$. The ‘stationary point’ of a corresponding fixed-boundary variational problem is characterized by a more general Euler-Lagrange system (see [BGG03, Section 1.2.2]). It is this latter notion that we have in mind when we refer to Euler-Lagrange systems.

Theoretically, the inverse problem:

Given a hyperbolic Monge-Ampère system with $S_{2}=\bm{0}$, can one construct an associated Lagrangian $2$-form $\Lambda$ in some local coordinates?

is solved in [BGG03, Theorems 1.2 and 2.2]. Here we only outline the steps of a construction and present some examples.

Start with a hyperbolic Monge-Ampère system with $S_{2}=\bm{0}$.

1. Step 1.

Find a $1$-adapted coframing $\bm{\omega}=(\omega^{0},\ldots,\omega^{4})$; 2. Step 2.

Compute the $1$-form $\phi_{0}$ in (19), which is determined, using $\bm{\omega}$; 3. Step 3.

Since, $S_{2}=\bm{0}$, $\phi_{0}$ must be closed; thus, find a function $\lambda$ such that $\mathrm{d}\lambda=2\lambda\phi_{0}$; 4. Step 4.

It would then follow that $\Pi:=\lambda\omega^{0}\wedge(\omega^{1}\wedge\omega^{2}-\omega^{3}\wedge\omega^{4})$ is closed; hence, it is a Poincaré-Cartan form. Now find161616This step can involve using the proof of the Poincaré Lemma. a $2$-form $\Lambda$ such that $\mathrm{d}\Lambda=\Pi$. This $\Lambda$ is a desired Lagrangian.

Now, for (66), (67) and the Monge-Ampère equation satisfying (65), we follow these steps to compute their representative Lagrangian 2-forms and summarize intermediate expressions in the following tables.

[TABLE]

[TABLE]

For the Monge-Ampère equation satisfying (65), we record only $\lambda$ and $\Lambda$, since the expressions of $\phi_{0}$ and $\Pi$ are rather complicated.

[TABLE]

In the above,

[TABLE]

Finally, we remind the reader that $\Lambda$ is not unique in the sense that one can add to it any exact $2$-form and any $2$-form in the contact ideal generated by $\mathrm{d}z-p\mathrm{d}x-q\mathrm{d}y$. For details, see [BGG03, Section 1.1].