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

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 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}.

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.

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.

2002

In the paper we investigate the theory of quantum optical systems. As an application we integrate and describe the quantum optical systems which are generically related to the classical orthogonal polynomials. The family of coherent states related to these systems is constructed and described. Some applications are also presented. 1

Symmetry, Integrability and Geometry: Methods and Applications, 2012

The standard binary Darboux transformation is investigated and is used to obtain quasi-Grammian multisoliton solutions of the generalized coupled dispersionless integrable system.

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

2002

In the paper we investigate the theory of quantum optical systems. As an application we integrate and describe the quantum optical systems which are generically related to the classical orthogonal polynomials. The family of coherent states related to these systems is constructed and described. Some applications are also presented.

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) = (

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.