Multi-Dimensional Conservation Laws and Integrable Systems

TL;DR

This paper introduces a new property of two-dimensional integrable systems involving infinitely many local three-dimensional conservation laws, and constructs such laws for key integrable models, also developing a new computation method.

Contribution

It presents a novel property of integrable systems, constructs new conservation laws for key models, and introduces a new method for computing local conservation laws in three-dimensional integrable systems.

Findings

Constructed infinitely many 3D conservation laws for KdV and Benney systems.

Developed a new method for computing conservation laws in 3D integrable systems.

Identified new quasi-local conservation laws and extended analysis to 4D laws.

Abstract

In this paper we introduce a new property of two-dimensional integrable systems -- existence of infinitely many local three-dimensional conservation laws for pairs of integrable two-dimensional commuting flows. Infinitely many three-dimensional local conservation laws for the Korteweg de Vries pair of commuting flows and for the Benney commuting hydrodynamic chains are constructed. As a by-product we established a new method for computation of local conservation laws for three-dimensional integrable systems. The Mikhalev equation and the dispersionless limit of the Kadomtsev--Petviashvili equation are investigated. All known local and infinitely many new quasi-local three-dimensional conservation laws are presented. Also four-dimensional conservation laws are considered for couples of three-dimensional integrable quasilinear systems and for triples of corresponding hydrodynamic chains.

Equations

$$u_t^k=F^k(\mathbf{u},{\mathbf{u}}_x,{\mathbf{u}}_{xx},{\mathbf{u}}_{xxx},\dots ),k=1,\dots ,N$$

$$\bigl{(}h_{p}(\mathbf{u},\mathbf{u}_{x},\mathbf{u}_{xx},\mathbf{u}_{xxx},\ldots)\bigr{)}_{t}=\bigl{(}g_{p}(\mathbf{u},\mathbf{u}_{x},\mathbf{u}_{xx},\mathbf{u}_{xxx},\ldots)\bigr{)}_{x}.$$

$$u_t^k=F^k(\mathbf{u},{\mathbf{u}}_x,{\mathbf{u}}_{xx},{\mathbf{u}}_{xxx},\dots ),u_y^k=G^k(\mathbf{u},{\mathbf{u}}_x,{\mathbf{u}}_{xx},{\mathbf{u}}_{xxx},\dots )$$

$$\bigl{(}A_{q}(\mathbf{u},\mathbf{u}_{x},\mathbf{u}_{xx},\ldots)\bigr{)}_{y}+\bigl{(}B_{q}(\mathbf{u},\mathbf{u}_{x},\mathbf{u}_{xx},\ldots)\bigr{)}_{t}+\bigl{(}C_{q}(\mathbf{u},\mathbf{u}_{x},\mathbf{u}_{xx},\ldots)\bigr{)}_{x}=0.$$

$$u_t^k=F^k(\mathbf{u},{\mathbf{u}}_x,{\mathbf{u}}_{xx},\dots ),u_y^k=G^k(\mathbf{u},{\mathbf{u}}_x,{\mathbf{u}}_{xx},\dots ),u_z^k=Q^k(\mathbf{u},{\mathbf{u}}_x,{\mathbf{u}}_{xx},\dots )$$

$$\bigl{(}A_{q}(\mathbf{u},\mathbf{u}_{x},\ldots)\bigr{)}_{z}+\bigl{(}B_{q}(\mathbf{u},\mathbf{u}_{x},\ldots)\bigr{)}_{y}+\bigl{(}C_{q}(\mathbf{u},\mathbf{u}_{x},\ldots)\bigr{)}_{t}+\bigl{(}D_{q}(\mathbf{u},\mathbf{u}_{x},\ldots)\bigr{)}_{x}=0.$$

$$u_{t^m}^k=F_m^k(\mathbf{u},{\mathbf{u}}_x,{\mathbf{u}}_{xx},\dots ),k=1,2,\dots ,N,m=2,\dots ,M$$

$$\partial_{t_{m}}\bigl{(}A_{m,q}(\mathbf{u},\mathbf{u}_{x},\ldots)\bigr{)}=0,\text{ \ }m=1,\ldots,M,$$

$$u_t={\left(-\frac{3}{2}{u}^2+{ϵ}^2{u}_{xx}\right)}_x$$

$$4{ϵ}^2{\psi }_{xx}=(\lambda +2u)\psi ,{\psi }_t=(\lambda -u){\psi }_x+\frac{1}{2}{u}_x\psi .$$

$$p_{t}=\Bigl{(}(\lambda-u)p\Bigr{)}_{x},$$

$$\epsilon^{2}\bigl{(}p^{-1}(\lambda)\bigr{)}_{xxx}=(\lambda+2u)\bigl{(}p^{-1}(\lambda)\bigr{)}_{x}+p^{-1}(\lambda)u_{x}.$$

$${\psi }_1\to \exp \left(\frac{\sqrt{\lambda }}{2ϵ}x\right),{\psi }_2\to \exp \left(-\frac{\sqrt{\lambda }}{2ϵ}x\right).$$

$$p=1+\frac{{\sigma }_1}{\lambda }+\frac{{\sigma }_2}{{\lambda }^2}+\frac{{\sigma }_3}{{\lambda }^3}+\dots$$

$$\bigl{(}\sigma_{k}\bigr{)}_{t}=\Bigl{(}\sigma_{k+1}-\sigma_{1}\sigma_{k}\Bigr{)}_{x},$$

$$u_t={\left(-\frac{3}{2}{u}^2+{ϵ}^2{u}_{xx}\right)}_x$$

$$4{ϵ}^2{\psi }_{xx}=(\lambda +2u)\psi ,{\psi }_t=(\lambda -u){\psi }_x+\frac{1}{2}{u}_x\psi ,$$

$$u_y={\left(\frac{5}{2}{u}^3-\frac{5}{2}{ϵ}^2{u}_x^2-5{ϵ}^{2}u{u}_{xx}+{ϵ}^4{u}_{xxxx}\right)}_x$$

$$4{ϵ}^2{\psi }_{xx}=(\lambda +2u)\psi ,{\psi }_y=({\lambda }^2-\lambda u-v){\psi }_x+\frac{1}{2}(\lambda u_x+v_x)\psi .$$

$$v=-\frac{3}{2}{u}^2+{ϵ}^2{u}_{xx}.$$

$${\psi }_t=(\lambda -u){\psi }_x+\frac{1}{2}{u}_x\psi ,{\psi }_y=({\lambda }^2-\lambda u-v){\psi }_x+\frac{1}{2}(\lambda u_x+v_x)\psi$$

$$p_{t}=\Bigl{(}(\lambda-u)p\Bigr{)}_{x},\text{ \ }p_{y}=\Bigl{(}(\lambda^{2}-\lambda u-v)p\Bigr{)}_{x}.$$

$$p=1+\frac{{\sigma }_1}{\lambda }+\frac{{\sigma }_2}{{\lambda }^2}+\frac{{\sigma }_3}{{\lambda }^3}+\dots$$

$$p_{t}=\Bigl{(}(\lambda-u)p\Bigr{)}_{x},\text{ \ }p_{y}=\Bigl{(}(\lambda^{2}-\lambda u-v)p\Bigr{)}_{x}$$

$$\bigl{(}\sigma_{k}\bigr{)}_{t}=\Bigl{(}\sigma_{k+1}-\sigma_{1}\sigma_{k}\Bigr{)}_{x},\quad\bigl{(}\sigma_{k}\bigr{)}_{y}=\Bigl{(}\sigma_{k+2}-\sigma_{1}\sigma_{k+1}+\bigl{(}\sigma_{1}^{2}-\sigma_{2}\bigr{)}\sigma_{k}\Bigr{)}_{x},$$

$$u_t={\left(-\frac{3}{2}{u}^2+{ϵ}^2{u}_{xx}\right)}_x,u_y={\left(\frac{5}{2}{u}^3-\frac{5}{2}{ϵ}^2{u}_x^2-5{ϵ}^{2}u{u}_{xx}+{ϵ}^4{u}_{xxxx}\right)}_x$$

$$\bigl{(}A_{k}(\sigma_{1},\ldots,\sigma_{k})\bigr{)}_{y}+\bigl{(}B_{k}(\sigma_{1},\ldots,\sigma_{k+1})\bigr{)}_{t}+\bigl{(}C_{k}(\sigma_{1},\ldots,\sigma_{k+2})\bigr{)}_{x}=0.$$

$$\bigl{(}p(\lambda)p(\zeta)\bigr{)}_{y}+\Bigl{(}\bigl{(}\sigma_{1}-\lambda-\zeta\bigr{)}p(\lambda)p(\zeta)\Bigr{)}_{t}+\Bigl{(}\bigl{(}\sigma_{2}-(\lambda+\zeta)\sigma_{1}+\lambda\zeta\bigr{)}p(\lambda)p(\zeta)\Bigr{)}_{x}=0.$$

$$p(\lambda )=1+\frac{{\sigma }_1}{\lambda }+\frac{{\sigma }_2}{{\lambda }^2}+\frac{{\sigma }_3}{{\lambda }^3}+\dots$$

$$\displaystyle\bigl{(}\sigma_{1}^{2}\bigr{)}_{y}+\bigl{(}\sigma_{1}^{3}-2\sigma_{1}\sigma_{2}\bigr{)}_{t}+\bigl{(}\sigma_{2}^{2}-\sigma_{1}^{2}\sigma_{2}\bigr{)}_{x}=0,$$

Full text

1. 1 Introduction

2. 1.1 Example: The Korteweg de Vries Equation

3. 2 The Dispersionless Limit of the Kadomtsev–Petviashvili Equation

4. 2.1 Higher Three-Dimensional Hydrodynamic Conservation Laws

5. 2.2 Three-Dimensional Hydrodynamic Conservation Laws

6. 2.3 Hydrodynamic Reductions

7. 2.4 Four-Dimensional Local Conservation Laws

8. 3 Mikhalëv Equation

9. 3.1 Hydrodynamic Reductions

10. 3.2 Finite-Dimensional Reductions

11. 3.3 Four-Dimensional Local Conservation Laws

12. 4 Appendix

13. 5 Conclusion

14. Acknowledgements

dedicated to 70th birthday of Franco Magri

keywords: integrable system, Benney hydrodynamic chain, Kadomtsev–Petviashvili equation, Korteweg de Vries equation, three-dimensional conservation laws.

MSC: 35F20, 35F50, 35L10, 35L25, 35L40, 35L60, 35L65, 35L70, 35L72, 35L77, 35N10, 35Q51, 35Q53, 37K05, 37K10, 58J45;

PACS: 02.30.IK, 02.30.JR, 02.70.WZ.

1 Introduction

One of most important properties of two-dimensional integrable systems

[TABLE]

is an existence of infinitely many local two-dimensional conservation laws

[TABLE]

In this paper we observe that pairs of two-dimensional integrable commuting flows

[TABLE]

possess also infinitely many local three-dimensional conservation laws

[TABLE]

Moreover, obviously, triples of two-dimensional integrable commuting flows

[TABLE]

possess infinitely many local four-dimensional conservation laws

[TABLE]

So, we come to the Statement that any $M-1$ commuting integrable systems

[TABLE]

possess infinitely many local M-dimensional conservation laws

[TABLE]

where summation is supposed over $m$ and we identify $x=t_{1}$, $t=t_{2}$, $y=t_{3}$ etc.111Simultaneously, in the three-dimensional case: $A_{q}\equiv A_{3,q},B_{q}\equiv A_{2,q},C_{q}\equiv A_{1,q}$; in the four-dimensional case $A_{q}=A_{4,q},B_{q}\equiv A_{3,q},C_{q}\equiv A_{2,q},D_{q}\equiv A_{1,q}$, etc.

Our claim is that this is a universal property of two-dimensional integrable systems. Without loss of generality and for simplicity we restrict our consideration on two most remarkable cases: the dispersionless limit of the Kadomtsev–Petviashvili equation (also known as the Khokhlov–Zabolotzkaya equation in nonlinear acoustics or the Lin–Reissner–Tsien equation in aerodynamics) and the Mikhalëv equation [17]. These three-dimensional equations are connected with hydrodynamic chains and possess infinitely many two-dimensional reductions. Any of these reductions is an integrable system. Pairs of such two-dimensional integrable systems have infinitely many local three-dimensional conservation laws. Corresponding three-dimensional integrable quasilinear equations possess infinitely many quasilocal three-dimensional conservation laws. Our approach is based on construction of two-parametric families of generating equations of quasilocal three-dimensional conservation laws. Expansion of these generating equations with respect to these parameters allows to extract all (a finite number) local three-dimensional conservation laws and infinitely many quasilocal three-dimensional conservation laws. Of course, this construction is valid for any number of dimensions and for any integrable systems.

1.1 Example: The Korteweg de Vries Equation

The Korteweg de Vries equation ($\epsilon$ is an arbitrary constant)

[TABLE]

follows from the compatibility condition \bigl{(}\psi_{xx}\bigr{)}_{t}=\bigl{(}\psi_{t}\bigr{)}_{xx}, where

[TABLE]

Introducing the function p(\lambda)=\bigl{(}\psi_{1}\psi_{2}\bigr{)}^{-1}, where $\psi_{1}$ and $\psi_{2}$ are two functionally independent solutions of the first equation in (1), the second equation in (1) transforms to the conservative form

[TABLE]

while the first equation in (1) becomes

[TABLE]

If $\lambda\rightarrow\infty$, one can choose solutions $\psi_{1}$ and $\psi_{2}$ with the following asymptotic behaviour

[TABLE]

This means that the function p(\lambda)=\bigl{(}\psi_{1}\psi_{2}\bigr{)}^{-1} has the following expansion

[TABLE]

Substitution this series into (2) leads to infinitely many local two-dimensional conservation laws

[TABLE]

where $u=\sigma_{1}$ and all conservation law densities $\sigma_{k}(u,u_{x},u_{xx},...)$ can be found from (3).

Now we consider simultaneously the Korteweg de Vries equation

[TABLE]

determined by the compatibility condition \bigl{(}\psi_{xx}\bigr{)}_{t}=\bigl{(}\psi_{t}\bigr{)}_{xx}, where

[TABLE]

and its first commuting flow

[TABLE]

determined by the compatibility condition \bigl{(}\psi_{xx}\bigr{)}_{y}=\bigl{(}\psi_{y}\bigr{)}_{xx}, where

[TABLE]

Here

[TABLE]

Again introducing the function p(\lambda)=\bigl{(}\psi_{1}\psi_{2}\bigr{)}^{-1}, where $\psi_{1}$ and $\psi_{2}$ are two functionally independent solutions of the first equation in (1), linear equations

[TABLE]

take the conservative form

[TABLE]

Substitution the expansion

[TABLE]

into generating equations of local two-dimensional conservation laws

[TABLE]

leads to infinitely many local two-dimensional conservation laws

[TABLE]

where $u=\sigma_{1}$ and $v=\sigma_{2}-\sigma_{1}^{2}$.

Theorem: The Korteweg de Vries pair of commuting flows

[TABLE]

possesses infinitely many local three-dimensional conservation laws

[TABLE]

Indeed, below we prove that the generating equation of these local three-dimensional conservation laws is

[TABLE]

Substitution the expansion ($\lambda\rightarrow\infty$)

[TABLE]

yields infinitely many local three-dimensional conservation laws

[TABLE]

where $\sigma_{k}$ determined recursively from (3) have the following form

[TABLE]

The structure of the paper is as follows. In Section 2 we investigate the dispersionless limit of the Kadomtsev–Petviashvili equation, the Benney pair of commuting hydrodynamic chains and their hydrodynamic reductions. We found local three-dimensional conservation laws for these integrable systems and infinitely many quasilocal three-dimensional conservation laws for the dispersionless limit of the Kadomtsev–Petviashvili equation. In Subsection 2.1 we rewrite all local three-dimensional conservation laws for the dispersionless limit of the Kadomtsev–Petviashvili equation as three-dimensional hydrodynamic conservation laws for the Benney pair of commuting hydrodynamic chains. In Subsection 2.2 we construct a generating equation of three-dimensional hydrodynamic conservation laws for the Benney pair of commuting hydrodynamic chains. We show that these conservation laws simultaneously are quasilocal three-dimensional conservation laws for the dispersionless limit of the Kadomtsev–Petviashvili equation. In Subsection 2.3 we proved that $N$ component hydrodynamic reductions of the Benney pair of commuting flows (and simultaneously of the dispersionless limit of the Kadomtsev–Petviashvili equation) possess $2N$ infinite series of three-dimensional hydrodynamic conservation laws. In Subsection 2.4 we demonstrate the dispersionless limit of the Kadomtsev–Petviashvili equation and its first commuting flow have infinitely many quasilocal four-dimensional conservation laws. In Section 3 we construct a generating equation of quasilocal three-dimensional conservation laws for the Mikhalëv equation. In Subsection 3.1 we present a family of three-dimensional hydrodynamic conservation laws parameterised by $2N$ arbitrary functions of a single variable for the linearly-degenerate hydrodynamic reductions. In Subsection 3.2 we show that $M$ commuting finite-dimensional systems have $M$-dimensional hydrodynamic conservation laws. In Subsection 3.3 we briefly discuss construction of four-dimensional hydrodynamic conservation laws for the Mikhalëv triple of hydrodynamic chains. In the Appendix 4 we carefully derive the over-determined system in involution, whose solution determines the generating function of three-dimensional hydrodynamic conservation law densities for the Benney pair of commuting hydrodynamic chains. Finally in Conclusion 5 we discuss open problems.

2 The Dispersionless Limit of the Kadomtsev–Petviashvili Equation

The remarkable dispersionless Kadomtsev–Petviashvili equation (see, for instance, [13]) can be written in three distinguish forms

[TABLE]

where $u=\psi_{x}=\Omega_{xx}$. Introducing the auxiliary field variable $v$ such that $u_{t}=v_{x}$, the first quasilinear equation of second order becomes the quasilinear system of two equations

[TABLE]

which possesses three hydrodynamic conservation laws (see [8])

[TABLE]

The dispersionless Kadomtsev–Petviashvili equation written in the potential form

[TABLE]

has four local conservation laws (see [5])

[TABLE]

whose densities and fluxes depend on $\psi_{x},\psi_{t},\psi_{y}$ only. Introducing hydrodynamic field variables $u=\psi_{x},v=\psi_{t}$ and $w=\psi_{y}$ one can see that the first two above conservation laws coincide with two first conservation laws of (4), while two others are new:

[TABLE]

The dispersionless Kadomtsev–Petviashvili equation written in the double potential form

[TABLE]

possesses already ten local conservation laws (see details in the Appendix A)

[TABLE]

where the conservation law densities $A_{k}$ and fluxes $B_{k},C_{k}$ depend on second order derivatives of a single function $\Omega$ only. Here we present a list of all particular solutions $A_{k},B_{m}$ and $C_{n}$:

[TABLE]

[TABLE]

Remark: Taking into account that $\psi=\Omega_{x}$, one can see that first above three dimensional conservation laws coincide with (6), (7), (8), while differentiation (9) with respect to $x$ is precisely (5).

Below we show that the dispersionless limit of the Kadomtsev–Petviashvili equation (9) possesses infinitely many local three-dimensional conservation laws \bigl{(}A_{k}\bigr{)}_{y}+\bigl{(}B_{k}\bigr{)}_{t}+\bigl{(}C_{k}\bigr{)}_{x}=0, whose conservation law densities $A_{k}$ and fluxes $B_{k},C_{k}$ depend on second order derivatives of a single function $\Omega$ with respect to all higher time variables.

2.1 Higher Three-Dimensional Hydrodynamic Conservation Laws

The dispersionless Kadomtsev–Petviashvili equation is an integrable three-dimensional quasilinear equation, which follows from the compatibility condition \bigl{(}p_{t}\,\bigr{)}_{y}=\bigl{(}p_{y}\bigr{)}_{t}, where

[TABLE]

The substitution

[TABLE]

into (11) leads to following consequences: $H_{0}=\Omega_{xx},H_{1}=\Omega_{xt},H_{2}=\Omega_{xy},$

[TABLE]

and to two commuting infinite series of two-dimensional conservation laws:

[TABLE]

[TABLE]

Taking into account (see (13))

[TABLE]

all above ten three-dimensional conservation laws can be written in the hydrodynamic type form, i.e. their densities $A_{k}$ and fluxes $B_{k},C_{k}$ depend on $H_{m}$ only:

[TABLE]

[TABLE]

So, all these three-dimensional conservation laws can be presented in the following form

[TABLE]

It is well known, that the Benney hydrodynamic chain (see [2], [10], [14], [25], [26])

[TABLE]

can be written in the conservative form (see, for instance, [22])

[TABLE]

where all two-dimensional conservation law densities $H_{k}$ are polynomials with respect to moments $A^{m}$, and can be found by substitution of inverse expansion

[TABLE]

into (12). Then H_{0}=A^{0},\,H_{1}=A^{1},\,H_{2}=A^{2}+\bigl{(}A^{0}\bigr{)}^{2},\,H_{3}=A^{3}+3A^{0}A^{1}, etc. Benney hydrodynamic chain written in conservative form (15) is nothing but precisely the first series of two-dimensional conservation laws in (14). Also one can prove in a similar way that the Benney pair of commuting hydrodynamic chains

[TABLE]

[TABLE]

can be written in conservative form (14), i.e. the second series of two-dimensional conservation laws in (14) is a conservative form for the first commuting flow (18) to Benney hydrodynamic chain (17).

Now we prove that this Benney pair of commuting hydrodynamic chains possesses infinitely many such three-dimensional hydrodynamic conservation laws.

2.2 Three-Dimensional Hydrodynamic Conservation Laws

The above ten three-dimensional hydrodynamic conservation laws can be separated according to the dependence on a highest density $H_{k}$, i.e. we have: one conservation law

[TABLE]

two conservation laws

[TABLE]

three conservation laws

[TABLE]

four conservation laws

[TABLE]

This situation is very unusual in comparison with two-dimensional conservation laws in the theory of two-dimensional integrable hydrodynamic chains, where just one two-dimensional conservation law

[TABLE]

exists for every index $k$. The number of three-dimensional hydrodynamic conservation laws

[TABLE]

is proportional to the index $k$. This means that the generating function of conservation law densities $A_{k}(H_{0},H_{1},\ldots,H_{k})$ cannot depend on a sole parameter only in a three-dimensional case.

Theorem: The generating equation of three-dimensional conservation laws

[TABLE]

for the dispersionless Kadomtsev–Petviashvili equation depends on two arbitrary parameters, i.e.

[TABLE]

[TABLE]

where $p(\lambda)\equiv p(\lambda;H_{0},H_{1},...)$ is a one-parametric generating function of two-dimensional conservation law densities $H_{k}$, and corresponding generating equations of two-dimensional conservation laws are

[TABLE]

Proof: One can look for a generating equation of three-dimensional hydrodynamic conservation laws in the following ansatz

[TABLE]

where we denoted $p\equiv p(\lambda)$ and $q\equiv p(\zeta)$. Then one can express all first derivatives $C_{p}\equiv\partial C/\partial p$, $C_{q}\equiv\partial C/\partial q$ and $C_{0}~{}\equiv~{}\partial C/\partial H_{0}$ via first derivatives of the functions $A$ and $B$, i.e.

[TABLE]

The compatibility conditions222Here \bigl{(}C_{p}\bigr{)}_{q}=C_{pq}\equiv\partial^{2}C/\partial p\partial q etc.

[TABLE]

lead to second derivatives of the function $B$ expressed via derivatives of the function $A$, i.e.

[TABLE]

The compatibility conditions333Here \bigl{(}B_{0p}\bigr{)}_{q}=B_{0pq}\equiv\partial^{3}C/\partial p\partial q\partial H_{0} etc.

[TABLE]

lead to the system in involution:

[TABLE]

This system can be integrated immediately, i.e.

[TABLE]

This means that $A_{pq}=p-q$ up to an arbitrary factor. Then (20) reduces to the form

[TABLE]

whose solution is $A=-(p-q)^{3}/6$. The factor $-1/6$ is inessential. So, one can fix $A=(p-q)^{3}$. The functions $B$ and $C$ can be found in quadratures. The Theorem is proved.

Remark: One can look for generating functions of three-dimensional hydrodynamic conservation law densities in the infinitely many various forms: $a_{k}(p,H_{0},H_{1},\ldots,H_{k})$, i.e. $a_{0}(p,H_{0})$, $a_{1}(p,H_{0},H_{1})$, $a_{2}(p,H_{0},H_{1},H_{2})$ etc. Similar computations lead to corresponding generating equations of three-dimensional hydrodynamic conservation laws are

[TABLE]

Alternatively one can expand (19) with respect to the parameter $\zeta$. Such an expansion of (19) leads to linear combinations of above generating equations of three-dimensional conservation laws with two-dimensional conservation laws (see (11)).

On the other side, the dispersionless limit of the Kadomtsev–Petviashvili equation belongs to the integrable hierarchy (see [20]), whose two-dimensional conservation law densities $H_{k}$ are nothing but second order derivatives of a sole function $\Omega$ with respect to higher time variables $t_{k}$, where we denoted $x=t_{0},t=t_{1}$ and $y=t_{2}$. Thus, taking into account that

[TABLE]

all three-dimensional conservation law densities and corresponding fluxes can be expressed in terms of $\Omega_{0k}$ only.

Definition: We call densities and fluxes of conservation laws for the dispersionless limit of the Kadomtsev–Petviashvili equation quasilocal, when they just depend on derivatives of a single function $\Omega$ with respect to higher time variables.

So, the dispersionless limit of the Kadomtsev–Petviashvili equation (9), indeed, possesses infinitely many quasilocal three-dimensional conservation laws \bigl{(}A_{k}\bigr{)}_{y}+\bigl{(}B_{k}\bigr{)}_{t}+\bigl{(}C_{k}\bigr{)}_{x}=0, whose densities $A_{k}$ and fluxes $B_{k},C_{k}$ depend on second order derivatives of a single function $\Omega$.

2.3 Hydrodynamic Reductions

Here we consider two-dimensional hydrodynamic reductions of the dispersionless limit of the Kadomtsev–Petviashvili equation and show that pairs of commuting two-dimensional hydrodynamic reductions possess infinitely many local three-dimensional hydrodynamic conservation laws.

J. Gibbons and S.P. Tsarev proved (see detail in [11], see also [12]) that the dispersionless limit of the Kadomtsev–Petviashvili equation has infinitely many $N$ component two-dimensional hydrodynamic reductions

[TABLE]

where the functions444In this Subsection we use the notation $A^{k}$ for moments following D.J. Benney, who introduced them in his seminal paper [2]. So we would like emphasise the difference between moment $A^{k}$ and three-dimensional conservation law densities $A_{k}$ in this article. $A^{0}(\mathbf{u})$ and $A^{1}(\mathbf{u})$ are not arbitrary and can be found by substitution the ansatz $A^{k}(\mathbf{u})$ into Benney pair of commuting hydrodynamic chains (17), (18), simultaneously. All these functions $A^{k}(\mathbf{u})$ can be found in quadratures, except the function $A^{0}(\mathbf{u})$ satisfying the nonlinear Gibbons–Tsarev system (see [21], [22])

[TABLE]

where we denote $\partial_{k}\equiv\partial/\partial A^{k}$ and the shift operator is $\delta=\sum\limits_{m}\partial_{m}$.

Without loss of generality we consider the so-called waterbag reduction (see also, for instance, [3], [11])

[TABLE]

where

[TABLE]

and $\epsilon_{k}$ are arbitrary constants. Substituting these expressions into (16), one can immediately find that the waterbag reduction is associated with the Riemann surface

[TABLE]

Thus, a pair of commuting two-dimensional hydrodynamic type systems (21) possesses infinitely many three-dimensional hydrodynamic conservation laws (19) where the dependence $p(\lambda)$ is inverse to (22). However, in comparison with the Benney pair of commuting hydrodynamic chains (14) any pair of $N$ component two-dimensional hydrodynamic reductions possesses $2N$ infinite series of three-dimensional hydrodynamic conservation laws. For instance, the generating function of two-dimensional hydrodynamic conservation law densities $p(\lambda)$ for the waterbag reduction has $N$ branches: $p(\lambda)=u^{k}+\lambda_{k}p_{1}^{(k)}+\lambda_{k}^{2}p_{2}^{(k)}+\ldots$, where $\lambda_{k}=e^{-\lambda/\epsilon_{k}}$ is a local parameter. Then all two-dimensional hydrodynamic conservation law densities $p_{m}^{(k)}$ can be found directly from (22). For example (see [21], [22]),

[TABLE]

Thus, the substitution $p(\zeta)=u^{k}+\zeta_{k}p_{1}^{(k)}+\zeta_{k}^{2}p_{2}^{(k)}+\ldots$ into (19) yields $N$ infinite series of one-parametric generating equations of three-dimensional hydrodynamic conservation laws:

[TABLE]

Then substitution $p(\lambda)=u^{m}+\lambda_{m}p_{1}^{(m)}+\lambda_{m}^{2}p_{2}^{(m)}+\ldots$ into these $N$ infinite series of one-parametric generating equations of three-dimensional hydrodynamic conservation laws leads to $2N$ infinite series of three-dimensional hydrodynamic conservation laws. For instance, first $N(N+1)/2$ three-dimensional hydrodynamic conservation laws are

[TABLE]

[TABLE]

[TABLE]

2.4 Four-Dimensional Local Conservation Laws

Here we again remind that our claim is that any $M-1$ commuting two-dimensional integrable systems possess infinitely many $M$-dimensional conservation laws. The case $M=3$ already is investigated. Now we consider the case $M=4$. Here we discuss four-dimensional local conservation laws for the first three two-dimensional commuting Benney hydrodynamic chains and simultaneously for two commuting three-dimensional quasilinear equations (of second and third orders, respectively). The dispersionless limit of the Kadomtsev–Petviashvili equation together with its first commuting flow

[TABLE]

have infinitely many four-dimensional conservation laws \bigl{(}A_{k}\bigr{)}_{z}+\bigl{(}B_{k}\bigr{)}_{y}+\bigl{(}C_{k}\bigr{)}_{t}+\bigl{(}D_{k}\bigr{)}_{x}=0, whose densities $A_{k}$ and fluxes $B_{k},C_{k},D_{k}$ depend on second derivatives of the sole function $\Omega$ with respect to all higher time variables.

Indeed, the compatibility conditions \bigl{(}p_{t}\bigr{)}_{y}=\bigl{(}p_{y}\bigr{)}_{t}, \bigl{(}p_{t}\bigr{)}_{z}=\bigl{(}p_{z}\bigr{)}_{t}, \bigl{(}p_{y}\bigr{)}_{z}=\bigl{(}p_{z}\bigr{)}_{y}, where

[TABLE]

lead to three algebraic equations connecting second order derivatives of the function $\Omega$, i.e.

[TABLE]

The compatibility condition \bigl{(}\Omega_{yt}\bigr{)}_{y}=\bigl{(}\Omega_{yy}\bigr{)}_{t} leads to (23). Then, substitution (12) into above generating equations of two-dimensional conservation laws (cf. (11)) leads to the three Benney commuting hydrodynamic chains:

[TABLE]

where we denoted $H_{0}=\Omega_{xx}$, $H_{1}=\Omega_{xt},$ $H_{2}=\Omega_{xy}$, $H_{3}=\Omega_{xz}$ and (cf. (13) and (24))

[TABLE]

Now one can look for a generating equation of four-dimensional hydrodynamic conservation laws $A_{z}+B_{y}+C_{t}+D_{x}=0$, where

[TABLE]

As in the previous (three-dimensional) case, these functions $A,B,C,D$ can be found explicitly. However, corresponding expressions are huge as well as four-dimensional conservation laws. So, we present here just the simplest example:

[TABLE]

3 Mikhalëv Equation

In this Section we consider another remarkable three-dimensional quasilinear equation of second order. The Mikhalëv equation (see [17])

[TABLE]

has global solutions in comparison with the dispersionless limit of the Kadomtsev–Petviashvili equation (see detail in [9], [23]). This is a linearly degenerate three-dimensional equation, which possesses infinitely many hydrodynamic and dispersive two-dimensional integrable reductions (see also [1], [16], [24]). Its simplest two-dimensional dispersive reduction is the Korteweg de Vries pair of commuting equations considered in the Introduction. So all multi-phase solutions of the Korteweg de Vries commuting pair are simultaneously global solutions of the Mikhalëv equation. Under the potential substitutions $u=w_{x}$ and $v=w_{t}$, Mikhalëv equation becomes the Mikhalëv system

[TABLE]

which follows from the compatibility condition \bigl{(}p_{t}\,\bigr{)}_{y}=\bigl{(}p_{y}\bigr{)}_{t}, where

[TABLE]

Also Mikhalëv system (26) can be derived from two commuting hydrodynamic chains

[TABLE]

which can be extracted from both generating equations of two-dimensional conservation laws (27) by substitution of the expansion

[TABLE]

In this case one can see that $u=\sigma_{1}$ and $v=\sigma_{2}-\sigma_{1}^{2}$. Then one can choose two first equations \bigl{(}\sigma_{1}\bigr{)}_{t}=\bigl{(}\sigma_{2}-\sigma_{1}^{2}\bigr{)}_{x}, \bigl{(}\sigma_{2}\bigr{)}_{t}=\bigl{(}\sigma_{3}-\sigma_{1}\sigma_{2}\bigr{)}_{x} of the first hydrodynamic chain and the first equation \bigl{(}\sigma_{1}\bigr{)}_{y}=\bigl{(}\sigma_{3}-2\sigma_{1}\sigma_{2}+\sigma_{1}^{3}\bigr{)}_{x} of the second hydrodynamic chain. Eliminating $\sigma_{3}$, finally we obtain Mikhalëv system (26).

One can repeat all computations presented in the previous Section. So we omit similar derivation and formulate

Theorem: The Mikhalëv pair of commuting hydrodynamic chains (28) possesses infinitely many three-dimensional hydrodynamic conservation laws

[TABLE]

Proof: We are looking for a two-parametric generating equation of three-dimensional conservation laws in the form

[TABLE]

where $p\equiv p(\lambda)$ and $q\equiv p(\zeta)$. Further reasons are the same as above. One can express the first derivatives $\partial C/\partial p\equiv C_{p}$, $\partial C/\partial q\equiv C_{q}$, $\partial C/\partial\sigma_{1}\equiv C_{1}$ and $\partial C/\partial\sigma_{2}\equiv C_{2}$ via first derivatives of the functions $A$ and $B$, i.e.

[TABLE]

The compatibility conditions

[TABLE]

lead to equations on second derivatives of the function $B$ expressed via derivatives of the function $A$, i.e.

[TABLE]

Their compatibility conditions lead to the simple system in involution

[TABLE]

Thus we found that the generating equation of three-dimensional hydrodynamic conservation laws for Mikhalëv pair of commuting hydrodynamic chains (28)

[TABLE]

also depends on two arbitrary parameters $\lambda,\zeta$. Then substitution (29) leads to infinitely many three-dimensional hydrodynamic conservation laws (30). The Theorem is proved.

Remark: In the limit $\zeta\rightarrow\lambda$, above two-parametric generating equation of three-dimensional hydrodynamic conservation laws reduces to the one parametric generating equation of three-dimensional hydrodynamic conservation laws

[TABLE]

Simultaneously (19) becomes

[TABLE]

Of course, these one-parametric generating equations cannot produce new three-dimensional conservation laws in comparison with two-parametric generating equation (31).

Taking into account (29) and expanding two-parametric generating equation (31) with respect to the parameter $\lambda$, one can obtain infinitely many one-parametric generating equations of three-dimensional conservation laws

[TABLE]

Expansion of these generating equations with respect to the parameter $\zeta$ yields corresponding three-dimensional hydrodynamic conservation laws (here we omit linear combinations of two-dimensional hydrodynamic conservation laws (28))

[TABLE]

One can consider higher commuting flows (see [23]), the first conservation law has the form (we remind that here $x=t_{1},t=t_{2},y=t_{3},z=t_{4}$)

[TABLE]

where the functions $a_{k}(\mathbf{\sigma})$ are connected with $\sigma_{m}$ via infinitely constraints

[TABLE]

Introducing the potential function $w$ such that555We remind that the potential function $w$ depends on infinitely time variables $t_{k}$ and Mikhalëv equation (25) is just a simplest three-dimensional quasilinear equation belonging to the corresponding linearly degenerate integrable hierarchy. $\sigma_{1}=w_{x}$ and $a_{k}=-\partial_{k}w$, we come to the conclusion that all other two-dimensional conservation law densities also can be expressed via first order derivatives of a single function $w$, for instance:

[TABLE]

Definition: We call densities and fluxes of conservation laws for the Mikhalëv equation quasilocal, when they just depend on derivatives of a single function $w$ with respect to higher time variables.

Thus, all above three-dimensional hydrodynamic conservation laws for the Mikhalëv pair of hydrodynamic chains simultaneously are three-dimensional quasilocal conservation laws of the Mikhalëv equation, whose densities and fluxes depend on first order derivatives of a sole function $w$, for instance:

[TABLE]

3.1 Hydrodynamic Reductions

A wide class of two-dimensional hydrodynamic reductions of Mikhalëv system (26) is the so called “$\epsilon$-systems”:

[TABLE]

where ($\epsilon_{k}$ are arbitrary constants)

[TABLE]

The generating function of two-dimensional conservation law densities $p(\lambda)\equiv p(\lambda,\mathbf{r}(x,t))$ can be found in quadratures directly from (27), i.e. (see [19])

[TABLE]

Now we consider the simplest case, i.e. all parameters $\epsilon_{k}=1$. The corresponding pair of linearly degenerate commuting hydrodynamic type systems (see (33)) possesses generating equation of three-dimensional hydrodynamic conservation laws (31). Then we can integrate this generating equation with respect to parameters $\lambda$ and $\zeta$ surrounding all simple poles $\lambda=r^{k}$, $\zeta=r^{m}$ and with arbitrary functions $R(\lambda)$, $L(\zeta)$, i.e. generating equation (31) takes the form $A_{y}+B_{t}+C_{x}=0$, where

[TABLE]

[TABLE]

These integrals can be found in explicit form:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $R_{k}(r^{k})$ and $L_{m}(r^{m})$ are arbitrary analytic functions. Thus, we found three-dimensional hydrodynamic conservation laws depended on $2N$ arbitrary functions of a single variable for pair of commuting hydrodynamic type systems (33).

3.2 Finite-Dimensional Reductions

Linearly degenerate hydrodynamic type systems (see (33), $\epsilon_{k}=1$)

[TABLE]

where

[TABLE]

have the general solution (see [7]):

[TABLE]

where $S_{k}(r^{k})$ are arbitrary functions. Indeed, one can compute first order derivatives $\partial x/\partial r^{k},\partial t/\partial r^{m},\partial y/\partial r^{n}$. Inverse expressions

[TABLE]

one can substitute back into (34) and obtain identities.

In the particular case ($E_{k}$ are arbitrary parameters)

[TABLE]

three commuting finite-dimensional systems (35) determine the $N$-phase solution of the Korteweg de Vries equation (see again [7], and formulae 2.3.12, 2.4.3 in [6], the first hydrodynamic type system in (34) can be found in [6] at page 139).

So, the pair of commuting finite-dimensional systems (see (35))

[TABLE]

has infinitely many two-dimensional conservation laws (see [18]; $T_{k}(r^{k})$ are arbitrary functions)

[TABLE]

while the triple of commuting finite-dimensional systems (35) has infinitely many three-dimensional conservation laws $A_{y}+B_{t}+C_{x}=0$, presented in the previous Subsection. Of course, instead of three commuting finite-dimensional systems (35), one can consider any number of such commuting finite-dimensional system (including higher commuting flows). Then corresponding multi-dimensional conservation laws can be easily found.

3.3 Four-Dimensional Local Conservation Laws

We already illustrated construction of generating equations of three-dimensional and four-dimensional quasilocal conservation laws for the dispersionless limit of the Kadomtsev–Petviashvili hierarchy. In this Section we constructed a generating equation of three-dimensional quasilocal conservation laws for Mikhalëv equation (25). So, our approach allows to construct generating equations of multi-dimensional quasilocal conservation laws for any integrable three-dimensional hierarchy. Even if such a hierarchy is not explicitly written. For instance, now we consider the dispersionless Lax “triple”

[TABLE]

where $u=\sigma_{1}$, $v=\sigma_{2}-\sigma_{1}^{2}$ and $s=\sigma_{1}^{3}-2\sigma_{1}\sigma_{2}+\sigma_{3}$. One can look for a generating equation of four-dimensional conservation laws $A_{z}+B_{y}+C_{t}+D_{x}=0$ in the ansatz

[TABLE]

[TABLE]

As in the three-dimensional case, these functions can be calculated explicitly. However, unlike to the case of the dispersionless limit of the Kadomtsev–Petviashvili equation, the generating equation has the fairly simple form

[TABLE]

The first non-trivial four-dimensional hydrodynamic conservation law has the following form

[TABLE]

Of course, one can substitute expansion (29) into (36) to extract first three Mikhalëv hydrodynamic chains, check the compatibility conditions \bigl{(}p_{t}\,\bigr{)}_{y}=\bigl{(}p_{y}\bigr{)}_{t}, \bigl{(}p_{t}\,\bigr{)}_{z}=\bigl{(}p_{z}\bigr{)}_{t}, \bigl{(}p_{z}\,\bigr{)}_{y}=\bigl{(}p_{y}\bigr{)}_{z}, and derive a corresponding linearly-degenerate system belonging to the Mikhalëv hierarchy. However, this linearly degenerate system also can be obtained directly from the above four-dimensional conservation law, if to remember that all functions $\sigma_{k}$ can be expressed from (32) as polynomials with respect to first derivatives of the common function $w$.

4 Appendix

In this section we compute densities666The ansatz for densities and fluxes does not depend on $\Omega_{tt}$ due to the algebraic relationship between second order derivatives in (9). $A_{k}(\Omega_{xx},\Omega_{xt},\Omega_{xy},\Omega_{yt},\Omega_{yy})$ and corresponding fluxes $B_{k}(\Omega_{xx},\Omega_{xt},\Omega_{xy},\Omega_{yt},\Omega_{yy})$, $C_{k}(\Omega_{xx},\Omega_{xt},\Omega_{xy},\Omega_{yt},\Omega_{yy})$ of local three-dimensional conservation laws (10) for the dispersionless limit of the Kadomtsev–Petviashvili equation written in algebraic form (9):

[TABLE]

As usual we rewrite the three-dimensional conservation law $A_{y}+B_{t}+C_{x}=0$ in the expanded form

[TABLE]

Taking into account differential consequences of the dispersionless limit of the Kadomtsev–Petviashvili equation

[TABLE]

and equating factors of all third derivatives of function $\Omega$ to zero we obtain the system

[TABLE]

Here the following notation is used $u=\Omega_{xx},v=\Omega_{xt},w=\Omega_{xy},r=\Omega_{yt},s=\Omega_{yy}$. The last equation in (37) implies that $A=A(u,v,w,r)$. Now we have to check compatibility conditions

[TABLE]

They imply a set of equations on second derivatives of function $B$. Taking into account differential consequences of equation $B_{s}+A_{r}=0$ from  (37), one can obtain the following system:

[TABLE]

[TABLE]

Compatibility conditions \bigl{(}B_{uu}\bigr{)}_{v}=\bigl{(}B_{uv}\bigr{)}_{u} etc. lead to the linear system in involution on third derivatives of the function $A(u,v,w,r)$:

[TABLE]

[TABLE]

Integration of this linear system in partial derivatives is elementary, because the most part of third derivatives vanishes. Straightforward computation yields the list of local three-dimensional conservation laws presented at the beginning of Section 2.

5 Conclusion

In this paper we established a new phenomenon in the theory of integrable systems. We discovered a new property: an existence of infinitely many local three-dimensional conservation laws for pairs of commuting two-dimensional integrable systems. We considered two remarkable three-dimensional integrable quasilinear systems of second order well known as the dispersionless limit of the Kadomtsev–Petviashvili equation and the Mikhalëv equation. We show that their two-dimensional hydrodynamic reductions (multi-component quasilinear systems of first order) have such local three-dimensional hydrodynamic conservation laws. We constructed two-parametric generating equations of three-dimensional hydrodynamic conservation laws for these hydrodynamic reductions and for corresponding hydrodynamic chains. The Mikhalëv equation also possesses infinitely many two-dimensional dispersive reductions. Also in this paper we considered the Korteweg de Vries pair of commuting flows as a simplest dispersive reduction of the Mikhalëv equation. Infinitely many local three-dimensional conservation laws were found for this pair of commuting dispersive equations.

For simplicity we restricted our consideration to pairs of commuting flows, whose Lax pairs are well known. Derivation of local three-dimensional conservation laws and corresponding generating equations is based on the theory of over-determined systems. Thus, one can consider any commuting pair of two-dimensional integrable systems and find corresponding local three-dimensional conservation laws.

Our motivation in this particular research was based on two important reasons. In the theory of shock waves discontinuities in a three-dimensional case can be interpreted as discontinuities of commuting two-dimensional reductions, which possess infinitely many local two-dimensional and three-dimensional conservation laws. Application of the Whitham averaging approach to three-dimensional integrable dispersive systems leads to three-dimensional quasilinear systems of first order, whose two-dimensional hydrodynamic reductions possess infinitely many two-dimensional and three-dimensional hydrodynamic conservation laws. We believe that our observation is a first step in derivation of infinitely many three-dimensional hydrodynamic conservation laws for such averaged systems. This problem should be investigated in a separate publication.

Our first computations of three-dimensional hydrodynamic conservation laws for the dispersionless limit of the Kadomtsev–Petviashvili equation show that the concept of local conservation laws can be extended to quasilocal conservation laws for three-dimensional integrable systems. At the first stage of our investigation (see details in [15]) we were able to prove existence of infinitely many such quasilocal conservation laws for the dispersionless limit of the Kadomtsev–Petviashvili equation, reducing corresponding computations to infinitely many over-determined systems in involutions, whose general solutions depend on arbitrary constants only. Instead of infinitely many particular calculations, in this paper we solved just one such an over-determined system, whose general solution determines a generating function of three-dimensional quasilocal conservation laws. This means that computation of infinitely many these conservation laws is reduced to the expansions of generating equations with respect to two arbitrary parameters.

The well known fact is that any integrable hydrodynamic chain possesses just one infinite series of two-dimensional hydrodynamic conservation laws, while its $N$ component hydrodynamic reductions have $N$ infinite series of two-dimensional hydrodynamic conservation laws. In this paper we show that pairs of commuting integrable hydrodynamic chains possess two-parametric family of three-dimensional hydrodynamic conservation laws, while corresponding $N$ component hydrodynamic reductions have $2N$ infinite series of three-dimensional hydrodynamic conservation laws. In a separate paper we are going to continue investigate other properties of multi-dimensional hydrodynamic conservation laws for families of two-dimensional commuting semi-Hamiltonian hydrodynamic type systems.

Acknowledgements

ZVM would like to thank A.A. Cherevko, E.V. Ferapontov and N.I. Makarenko for helpful conversations. MVP is grateful also to V.E. Adler, A.V. Aksënov, G.A. Alekseev, L.V. Bogdanov, I.S. Krasil’shchik, A.G. Kulikovskii, A.Ya. Maltsev, O.I. Morozov, D.V. Treschev, A.M. Verbovetsky for very important comments, remarks and advices. This work was partially supported by the Russian Science Foundation (grant No. 15-11-20013).