Integrable (3+1)-dimensional system with an algebraic Lax pair

TL;DR

This paper introduces the first known integrable (3+1)-dimensional dispersionless system with a nonisospectral algebraic Lax pair, expanding the class of such integrable systems beyond previously known rational cases.

Contribution

It presents a novel integrable (3+1)-dimensional system with an algebraic Lax pair, demonstrating greater diversity in nonisospectral Lax pairs for such systems.

Findings

First example of algebraic nonisospectral Lax pair in (3+1)D

Expands understanding of integrable dispersionless systems

Shows algebraic dependence broadens integrable system classes

Abstract

We present a first example of an integrable (3+1)-dimensional dispersionless system with nonisospectral Lax pair involving algebraic, rather than rational, dependence on the spectral parameter, thus showing that the class of integrable (3+1)-dimensional dispersionless systems with nonisospectral Lax pairs is significantly more diverse than it appeared before. The Lax pair in question is of the type recently introduced in [A. Sergyeyev, Lett. Math. Phys. 108 (2018), no. 2, 359-376, arXiv:1401.2122 ].

Equations

$$A_0(\mathbf{u}){\mathbf{u}}_t+A_1(\mathbf{u}){\mathbf{u}}_x+A_2(\mathbf{u}){\mathbf{u}}_y+A_3(\mathbf{u}){\mathbf{u}}_z=0,$$

$${\chi }_y=X_f(\chi ),{\chi }_t=X_g(\chi ),$$

$$X_h=h_p{\partial }_x+(p{h}_z-h_x){\partial }_p+(h-p{h}_p){\partial }_z$$

$$f=p^{n+1}+\sum _{j=0}^n{v}_j{p}^j,g=p^{m+1}+\frac{m}{n}{v}_n{p}^m+\sum _{k=0}^{m-1}{w}_k{p}^k$$

$$f=\sum _{j=1}^m{a}_i/(v_i-p),g=\sum _{k=1}^n{b}_k/(w_k-p)$$

$${\psi }_y={\psi }_{z}f({\psi }_x/{\psi }_z,\mathbf{u}),{\psi }_t={\psi }_{z}g({\psi }_x/{\psi }_z,\mathbf{u})$$

$$\begin{array}[]{rcl}a_{t}&=&\displaystyle\frac{1}{r^{2}-2rsa+2s^{2}b}\biggl{(}(4w(ra-sb)-vr)a_{x}+ra_{y}\\[11.38109pt] &&+\bigl{(}2w\left(2a(ra-sb)-rb\right)-ur\bigr{)}a_{z}\\[5.69054pt] &&+(vs-2wr)b_{x}-sb_{y}+(2w(sb-ra)+us)b_{z}\\[5.69054pt] &&+(r-sa)u_{x}+(ra-2sb)u_{z}+(2sb-ra)v_{x}\\[5.69054pt] &&+2(a(sb-ra)+rb)v_{z}\\[5.69054pt] &&+2(a(ar-sb)-rb)w_{x}+2\left(2a^{2}(ar-sb)-3rab+2sb^{2}\right)w_{z}\biggr{)},\\[14.22636pt] b_{t}&=&\displaystyle\frac{1}{r^{2}-2rsa+2s^{2}b}\biggl{(}2(2wr-vs)ba_{x}+2sba_{y}\\[11.38109pt] &&+2(2w(ra-sb)-us)ba_{z}\\[5.69054pt] &&+(2s(va-2wb)-vr)b_{x}+(r-2sa)b_{y}\\[5.69054pt] &&+(2(usa-wrb)-ur)b_{z}\\[5.69054pt] &&+(2s(b-a^{2})+ra)u_{x}+2(r-sa)bu_{z}\\[5.69054pt] &&-2(r-sa)bv_{x}-2(ra-2sb)bv_{z}\\[5.69054pt] &&+2(ra-2sb)bw_{x}+4(a(ra-sb)-rb)bw_{z}\biggr{)},\\[14.22636pt] r_{t}&=&\displaystyle\frac{1}{r^{2}-2rsa+2s^{2}b}\biggl{(}(vs-2wr)ra_{x}-rsa_{y}\\[11.38109pt] &&-(2w(ra-sb)-us)ra_{z}\\[5.69054pt] &&+(2wr-vs)sb_{x}+s^{2}b_{y}+(wr^{2}-us^{2})b_{z}\\[5.69054pt] &&+(sa-r)su_{x}+(2sb-ra)su_{z}+(r-sa)rv_{x}+(ra-2sb)rv_{z}\\[5.69054pt] &&+(2sb-ra)rw_{x}-2(a(ra-sb)-rb)rw_{z}\biggr{)},\\[14.22636pt] s_{t}&=&w_{x}+aw_{z}+wa_{z},\\ u_{t}&=&ar_{x}+2br_{z}-sb_{x},\\ v_{t}&=&r_{x}+ar_{z}+as_{x}+2bs_{z}-sa_{x}+sb_{z},\\ w_{t}&=&s_{x}+as_{z}+sa_{z}.\end{array}$$

$$\begin{array}[]{rcl}f&=&u+vp+wp^{2}+(r+sp)\sqrt{p^{2}+2ap+2b},\\[5.69054pt] g&=&\sqrt{p^{2}+2ap+2b},\end{array}$$

$$\begin{array}[]{rcl}\chi_{y}&=&\displaystyle\frac{1}{g}\biggl{(}\bigl{(}2sp^{2}+(r+3sa+2wg)p+ra+vg+2sb\bigr{)}\chi_{x}\\[14.22636pt] &&+\bigl{(}-sp^{3}-(wg+sa)p^{2}+pra+2rb+ug\bigr{)}\chi_{z}\\[5.69054pt] &&+\bigl{(}s_{z}p^{4}+(2as_{z}+sa_{z}+r_{z}+gw_{z}-s_{x})p^{3}\\[5.69054pt] &&+((v_{z}-w_{x})g+2bs_{z}+ra_{z}+sb_{z}-2as_{x}-sa_{x}-r_{x}+2ar_{z})p^{2}\\[5.69054pt] &&+((u_{z}-v_{x})g+rb_{z}-ra_{x}-sb_{x}-2bs_{x}-2r_{x}a+2br_{z})p\\[5.69054pt] &&-rb_{x}-gu_{x}-2br_{x}\bigr{)}\chi_{p}\biggr{)},\\[14.22636pt] \chi_{t}&=&\displaystyle\frac{1}{g}\biggl{(}(p+a)\chi_{x}+(ap+2b)\chi_{z}+(a_{z}p^{2}+p(b_{z}-a_{x})-b_{x})\chi_{p}\biggr{)},\end{array}$$

$$\begin{array}[]{rcl}\psi_{y}&=&u\psi_{z}+v\psi_{x}+w\psi_{x}^{2}/\psi_{z}+(r\psi_{z}+s\psi_{x})\sqrt{(\psi_{x}/\psi_{z})^{2}+2a\psi_{x}/\psi_{z}+2b},\\[8.53581pt] \psi_{t}&=&\psi_{z}\sqrt{(\psi_{x}/\psi_{z})^{2}+2a\psi_{x}/\psi_{z}+2b};\end{array}$$

$$f_t-g_y+\{f,g\}=0$$

$$\{f,g\}=f_p{g}_x-f_x{g}_p-p(f_p{g}_z-g_p{f}_z)+f{g}_z-g{f}_z.$$

Full text

Integrable (3+1)-dimensional system

with an algebraic Lax pair

A. Sergyeyev

Mathematical Institute, Silesian University in Opava,

Na Rybníčku 1, 746 01 Opava, Czech Republic

E-mail: Artur.Sergyeyev@math.slu.cz

(January 17, 2019)

We present a first example of an integrable (3+1)-dimensional dispersionless system with nonisospectral Lax pair involving algebraic, rather than rational, dependence on the spectral parameter, thus showing that the class of integrable (3+1)-dimensional dispersionless systems with nonisospectral Lax pairs is significantly more diverse than it appeared before. The Lax pair in question is of the type recently introduced in [A. Sergyeyev, Lett. Math. Phys. 108 (2018), no. 2, 359-376, arXiv:1401.2122].

Introduction

Integrable systems are well known to play an important role in modern mathematics and physics, cf. e.g. [1, 6, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 21, 23, 24, 25, 26]. According to Einstein’s general relativity our spacetime is four-dimensional, so the search for (3+1)-dimensional integrable systems, that is, integrable partial differential systems in four independent variables, is quite naturally of particular significance, see e.g. [9, 16, 19, 24]; also cf. e.g. [22] on pecularities of 4-manifolds. As discussed e.g. in [19], most of integrable (3+1)-dimensional systems known to date are dispersionless, which, roughly speaking, means that they can be written as first-order quasilinear homogeneous systems.

While a fairly large number of (3+1)-dimensional integrable systems is presently known, see e.g. [16, 18, 19] and references therein, the breadth of their class yet remains to be fully understood. In 2+1 dimensions there are many examples of dispersionless integrable systems whose nonisospectral Lax pairs involve highly sophisticated dependence on the spectral parameter, including, for example, Weierstrass $\wp$-functions, cf. e.g. [11], but, to the best of our knowledge, for all previously known examples of (3+1)-dimensional integrable dispersionless systems in finitely many dependent variables with nonisospectral Lax pairs the latter are polynomial or rational in the spectral parameter, which begets the question of whether more sophisticated nonisospectral Lax pairs could exist in 3+1 dimensions.

We answer this question in the affirmative by presenting in Section 2 below system (5) which is, as far as the present author is aware, the first example of an integrable (3+1)-dimensional dispersionless system in finitely many dependent variables with a nonisospectral Lax pair being algebraic, rather than merely rational, in the spectral parameter, which shows that nonisospectral dispersionless Lax pairs in 3+1 dimensions are significantly more diverse than it appeared before. The example in question is found within the framework of a new systematic construction for (3+1)-dimensional integrable systems related to contact geometry, see [19], the follow-up papers [4, 20], and a brief review in Section 1 below.

1 Preliminaries

Recall that a dispersionless or hydrodynamic-type system in four independent variables $x,y,z,t$ is, cf. e.g. [10, 11, 19, 21] and references therein, a system that can be written in the form

[TABLE]

where $\boldsymbol{u}=(u_{1},\dots,u_{N})^{\mathrm{T}}$ is an $N$-component vector of unknown functions of $x,y,z,t$, $A_{i}$ are $M\times N$-matrix-valued functions of $\boldsymbol{u}$, $M$ and $N$ are nonzero nonnegative integers such that $M\geqslant N$, and the superscript ${\mathrm{T}}$ indicates the transposed matrix.

Here and below all functions are assumed sufficiently smooth for all computations to make sense. This can be readily formalized using the language of differential algebra, cf. e.g. [7, 19], and references therein.

There exist [19] infinitely many integrable dispersionless systems of the general form (1) that admit, for suitable Lax functions $f=f(p,\boldsymbol{u})$ and $g=g(p,\boldsymbol{u})$, nonisospectral Lax pairs of the form introduced in [19] and intimately related to contact geometry,

[TABLE]

where $\chi=\chi(x,y,z,t,p)$, and $p$ is the so-called variable spectral parameter (note that $\boldsymbol{u}_{p}\equiv 0$).

For any $h=h(p,\boldsymbol{u})$ the operator $X_{h}$ is defined as

[TABLE]

and formally looks exactly like the contact vector field with a contact hamiltonian $h$ on a contact 3-manifold with local coordinates $x,z,p$ and contact one-form $dz+pdx$; cf. e.g. [3, 5] and references therein for more details on contact geometry.

In particular, for any natural $m$ and $n$ the pairs of Lax functions

[TABLE]

and

[TABLE]

yield [19, 20] (3+1)-dimensional integrable systems of the general form (1) with $M=N$ for $\boldsymbol{u}=(v_{0},\dots,v_{n},w_{0},\dots,\allowbreak w_{m-1})^{\mathrm{T}}$ and $\boldsymbol{u}=(a_{1},\dots,a_{m},b_{1},\dots,b_{n},v_{1},\dots,v_{m},w_{1},\dots,w_{n})^{\mathrm{T}}$ respectively.

Notice [19] that if $\boldsymbol{u}_{z}=0$ and $\chi_{z}=0$ then the Lax pairs (2) boil down to well-known (2+1)-dimensional Lax pairs involving Hamiltonian (rather than contact) vector fields, cf. e.g. [11, 26] and references therein for the associated (2+1)-dimensional integrable systems.

Proposition 1** ([19]).**

A system (1) admits a linear Lax pair of the form (2) if and only if it admits a nonlinear Lax pair for $\psi=\psi(x,y,z,t)$ of the form

[TABLE]

with the same functions $f$ and $g$ as in (2).

In closing note that systems of the form (4) belong to a broader class of multitime Hamilton–Jacobi systems, cf. e.g. [15] for more details on the latter.

2 New integrable system with algebraic Lax pair

Consider the following (3+1)-dimensional evolutionary system for $\boldsymbol{u}=(a,b,r,s,u,v,w)^{\mathrm{T}}$:

[TABLE]

Theorem 1**.**

The (3+1)-dimensional seven-component evolutionary system (5) is integrable since it admits a Lax pair (2) with algebraic Lax functions $f$ and $g$ given by

[TABLE]

that is,

[TABLE]

and a nonlinear Lax pair of the form (4) with $f$ and $g$ given by (6), that is,

[TABLE]

both of the Lax pairs (7) and (8) are expressed in terms of algebraic, rather than rational, functions.

Proof.

By definition, proving that (5) admits (7) and (8) amounts to proving that both (7) and (8) are compatible by virtue of (5).

First of all, by Proposition 1 compatibility of (7) by virtue of (5) implies the same for (8).

Next, by Proposition 1 of [19], in order to prove that (7) is compatible by virtue of (5), it suffices to show that for $f$ and $g$ given by (6) the system (5) implies the equation

[TABLE]

where the contact bracket $\{,\}$ is defined [19] as

[TABLE]

Substituting (6) into (9) we readily see that this is indeed the case, and the result follows. ∎

3 Conclusions

We have presented above a new integrable (3+1)-dimensional dispersionless evolutionary system (5) whose linear Lax pair (7) and nonlinear Lax pair (8) are expressed in terms of algebraic rather than rational functions.

To the best of our knowledge, this is the first example of an integrable (3+1)-dimensional dispersionless system in finitely many dependent variables admitting a nonisospectral Lax pair which is not rational in the variable spectral parameter. Indeed, as far as the present author is able to tell, for all previously known examples of integrable dispersionless (3+1)-dimensional systems in finitely many dependent variables with nonisospectral Lax pairs, including e.g. the equations for (anti-)self-dual four-dimensional conformal structures [10], the Dunajski equation, and systems (15), (17), (38), and (40) from [19], their Lax pairs have at most rational dependence on the variable spectral parameter.

We conjecture that (5) has no nontrivial linear or nonlinear Lax pairs written in terms of rational (rather than algebraic) functions.

In closing note that it could be of interest to study symmetries, conservation laws, Hamiltonian operators, and other related structures for (5) in spirit of [14, 17].

Acknowledgments

This research was supported in part by the Ministry of Education, Youth and Sports of the Czech Republic (MŠMT ČR) under RVO funding for IČ47813059, and by the Grant Agency of the Czech Republic (GA ČR) under grant P201/12/G028. The computations in the paper were mostly performed using the computer algebra package Jets [2] for Maple®.

I am pleased to thank E. Ferapontov, I.L. Freire, B. Kruglikov, H.V. Lê, R.O. Popovych, and R. Vitolo for stimulating discussions, C. Taubes for sending me a copy of [22], and the anonymous referees for useful suggestions.