From contact geometry to multidimensional integrable systems

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.

Letters in Mathematical Physics, 2018

We introduce a novel systematic construction for integrable (3+1)-dimensional dispersionless systems using nonisospectral Lax pairs that involve contact vector fields. In particular, we present new large classes of (3+1)-dimensional integrable dispersionless systems associated to the Lax pairs which are polynomial and rational in the spectral parameter.

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.

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.

Journal of Geometry and Physics, 2015

We consider Hamiltonian systems restricted to the hypersurfaces of contact type and obtain a partial version of the Arnold-Liouville theorem: the system not need to be integrable on the whole phase space, while the invariant hypersurface is foliated on an invariant Lagrangian tori. In the second part of the paper we consider contact systems with constraints. As an example, the Reeb flows on Brieskorn manifolds are considered. Contents 16 5. Note on K-contact manifolds 22 References 23

1998

Using the differential geometry of curves and surfaces, the L-equivalent soliton equations of the some (2+1) -dimensional integrable spin systems are found. These equations include the modified Novikov-Veselov, Kadomtsev-Petviashvili, Nizhnik-Novikov-Veselov and other equations. Some aspects of the connection between geometry and multidimensional soliton equations are discussed.

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.

A bi-Hamiltonian formulation for stationary flows of the KdV hierarchy is derived in an extended phase space. A map between stationary flows and restricted flows is constructed: in a case it connects the Henon-Heiles and the Garnier system. Moreover a new integrability scheme for Hamiltonian systems in their standard phase space is proposed. 4 2.1. Bi-Hamiltonian hierarchies and Gelfand-Dickey polynomials 4 2.2. The method of stationary flows 10 2.3. Example I: the bi-Hamiltonian structure of a Henon-Heiles system 16 2.4. The method of restricted flows 20 2.5. A map between stationary flows and restricted flows 21 2.6. Example II: the bi-Hamiltonian structure of the Garnier system 24 2.7. Example III: a map between the Henon-Heiles and the Garnier system 27 3. A new integrability structure 29 3.1. The reduced structures of Henon-Heiles and Garnier systems 29 3.2. A new integrability criterion 30 3.3. The integrability structure of Henon-Heiles and Garnier systems 33 4. A Henon-Heiles system with four degrees of freedom 35 4.1. The bi-Hamiltonian structure 35 4.2. The integrability structure 39 5. Concluding remarks 41 References 42

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 ].