New integrable (3+1)-dimensional systems and contact geometry

Nonlinear Systems and Their Remarkable Mathematical Structures, 2019

We review the recent approach to the construction of (3+1)-dimensional integrable dispersionless partial differential systems based on their contact Lax pairs and the related R-matrix theory for the Lie algebra of functions with respect to the contact bracket. We discuss various kinds of Lax representations for such systems, in particular, linear nonisospectral contact Lax pairs and nonlinear contact Lax pairs as well as the relations among the two. Finally, we present a large number of examples with finite and infinite number of dependent variables, as well as the reductions of these examples to lower-dimensional integrable dispersionless systems.

We introduce a new kind of nonlinear Lax-type representation, a contact Lax pair, related to the so-called contact, or Lagrange, bracket, and employ the former for the construction of a broad new class of (3+1)-dimensional integrable systems, thus showing that such systems are considerably less exceptional than it was hitherto believed. We further show that the (3+1)-dimensional integrable systems from this new class also admit a linear Lax representation with variable spectral parameter, and thus can be solved using the techniques like the inverse scattering transform, the dressing method, or the twistor approach. To illustrate our results, we construct inter alia a new (3+1)-dimensional integrable system with an arbitrary finite number of components. In one of the simplest special cases this system yields a (3+1)-dimensional integrable generalization of the dispersionless Kadomtsev–Petviashvili equation, also known as the Lin–Reissner–Tsien equation or (2+1)-dimensional Khokhlov–Zabolotskaya equation.

Boletín de la Sociedad Matemática Mexicana, 2025

Upon having presented a bird's eye view of history of integrable systems, we give a brief review of certain recent advances in the longstanding problem of search for partial differential systems in four independent variables that are integrable in the sense of soliton theory (such systems are known as integrable (3+1)-dimensional systems, or, in terms used in physics, classical integrable 4D field theories, in general non-relativistic and non-Lagrangian). Namely, we review a recent construction for a large new class of (3+1)-dimensional integrable systems with Lax pairs involving contact vector fields. This class contains two infinite families of such systems, thus establishing that there is significantly more integrable (3+1)-dimensional systems than it was believed for a long time. In fact, the construction under study yields (3+1)-dimensional integrable generalizations of many well-known dispersionless integrable (2+1)-dimensional systems like the dispersionless KP equation, as well as a first example of a (3+1)-dimensional integrable system with an algebraic, rather than rational, nonisospectral Lax pair. To demonstrate the versatility of the construction in question, we employ it here to produce novel integrable (3+1)-dimensional generalizations for the following (2+1)-dimensional integrable systems: dispersionless BKP, dispersionless asymmetric Nizhnik-Veselov-Novikov, dispersionless Gardner, and dispersionless modified KP equations, and the generalized Benney system.

Applied Mathematics Letters, 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 ].

Journal of Mathematical Sciences and Modelling

Our review is devoted to Lie-algebraic structures and integrability properties of an interesting class of nonlinear dynamical systems called the dispersionless heavenly equations, which were initiated by Plebański and later analyzed in a series of articles. The AKS-algebraic and related R-structure schemes are used to study the orbits of the corresponding co-adjoint actions, which are intimately related to the classical Lie-Poisson structures on them. It is demonstrated that their compatibility condition coincides with the corresponding heavenly equations under consideration. Moreover, all these equations originate in this way and can be represented as a Lax compatibility condition for specially constructed loop vector fields on the torus. The infinite hierarchy of conservations laws related to the heavenly equations is described, and its analytical structure connected with the Casimir invariants, is mentioned. In addition, typical examples of such equations, demonstrating in detail their integrability via the scheme devised herein, are presented. The relationship of a fascinating Lagrange-d'Alembert type mechanical interpretation of the devised integrability scheme with the Lax-Sato equations is also discussed. We pay a special attention to a generalization of the devised Lie-algebraic scheme to a case of loop Lie superalgebras of superconformal diffeomorphisms of the 1|N-dimensional supertorus. This scheme is applied to constructing the Lax-Sato integrable supersymmetric analogs of the Liouville and Mikhalev-Pavlov heavenly equation for every N ∈ N\{4; 5}.

Journal of Mathematical Analysis and Applications, 2018

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Highlights • An integrable discretization of the generalized coupled dispersionless integrable system (dGCD) system via Lax pair is presented. • A Darboux transformation is proposed for dGCD system. • Darboux matrix is defined in terms of quasideterminant. • Quasideterminant multisoliton solutions have been computed. • Explicit expressions of one and two soliton solutions have been computed.

2017

We present a new spectral problem, a generalization of the D-Kaup-Newell spectral problem, associated with the Lie algebra sl(2,R). Zero curvature equations furnish the soliton hierarchy. The trace identity produces the Hamiltonian structure for the hierarchy. Lastly, a reduction of the spectral problem is shown to have a different soliton hierarchy with a bi-Hamiltonian structure. The first major motivation of this dissertation is to present spectral problems that generate two soliton hierarchies with infinitely many commuting conservation laws and high-order symmetries, i.e., they are Liouville integrable. We use the soliton hierarchies and a non-seimisimple matrix loop Lie algebra in order to construct integrable couplings. An enlarged spectral problem is presented starting from a generalization of the D-Kaup-Newell spectral problem. Then the enlarged zero curvature equations are solved from a series of Lax pairs producing the desired integrable couplings. A reduction is made of ...

Central European Journal of Mathematics, 2007

The Hamiltonian representation for a hierarchy of Lax type equations on a dual space to the Lie algebra of integro-differential operators with matrix coefficients, extended by evolutions for eigenfunctions and adjoint eigenfunctions of the corresponding spectral problems, is obtained via some special Bäcklund transformation. The connection of this hierarchy with integrable by Lax twodimensional Davey-Stewartson type systems is studied.

Advances in Mathematical Physics, 2016

We study Lax triples (i.e., Lax representations consisting of three linear equations) associated with families of surfaces immersed in three-dimensional Euclidean space E 3. We begin with a natural integrable deformation of the principal chiral model. Then, we show that all deformations linear in the spectral parameter are trivial unless we admit Lax representations in a larger space. We present an explicit example of triply orthogonal systems with Lax representation in the group Spin(6). Finally, the obtained results are interpreted in the context of the soliton surfaces approach.

Filomat, 2012

In this paper we study the coupled integrable dispersionless system (CIDS), which arises in the analysis of several problems in applied mathematics and physics. Lie symmetry analysis is performed on CIDS and symmetry reductions and exact solutions with the aid of simplest equation method are obtained. In addition, the conservation laws of the CIDS are also derived using the multiplier (and homotopy) approach.