Multidimensional integrable systems from contact geometry

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.

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.

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.

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 ].

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

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.

2008

We give explicitly N-soliton solutions of a new (2 + 1) dimensional equation, φxt + φxxxz/4 + φxφxz + φxxφz/2 + ∂ −1 x φzzz/4 = 0. This equation is obtained by unifying two directional generalization of the KdV equation, composing the closed ring with the KP equation and Bogoyavlenskii-Schiff equation. We also find the Miura transformation which yields the same ring in the corresponding modified equations. Short title: LETTER TO THE EDITOR February 9, 2008 † [email protected] ‡ [email protected] ‖ [email protected] 2 The study of higher dimensional integrable system is one of the central themes in integrable systems. A typical example of higher dimensional integrable systems is to modify the Lax operators of a basic equation, in this letter the potential KdV(p-KdV) equation. The Lax pair of the p-KdV equation have the form L(x, t) = ∂ x + φx(x, t), (1) T (x, t) = (

The Hamiltonian equation provides us an alternate description of the basic physical laws of motion, which is used to be described by Newton's law. The research on Hamiltonian integrable systems is one of the most important topics in the theory of solitons. This article proposes a new hierarchy of integrable systems of 1 + 2 dimensions with its Hamiltonian form by following the residue approach of Fokas and Tu. The new hierarchy of integrable system is of fundamental interest in studying the Hamiltonian systems.

arXiv (Cornell University), 2016

The work is devoted to old and recent investigations of the classical M. A. Buhl problem of describing compatible linear vector field equations, its general M.G. Pfeiffer and modern Lax-Sato type special solutions. Eespecially we analyze the the related Lie-algebraic structures and integrability properties of a very interesting class of nonlinear dynamical systems called the dispersionless heavenly type 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 type equations under consideration. It is shown that 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 the very interesting Lagrange-d'Alembert type mechanical interpretation of the devised integrability scheme with the Lax-Sato equations is also discussed.

Quaternionic vector mKDV equations are derived from the Cartan structure equation in the symmetric space HPn = Sp(n+1)/Sp(1)◊Sp(n). The derivation of the soliton hierarchy utilizes a moving paralell frame and a Cartan connection 1-form ! related to the Cartan geometry on HPn modeled on (spn+1,sp1 ◊ spn). The integrability structure is shown to be geometrically encoded by a Poisson-Nijenhuis structure and a symplectic operator.