Matrix Lax Representation Research Articles

Local Yang–Baxter correspondences and set-theoretical solutions to the Zamolodchikov tetrahedron equation

We study tetrahedron maps, which are set-theoretical solutions to the Zamolodchikov tetrahedron equation, and their matrix Lax representations defined by the local Yang–Baxter equation. Sergeev (1998 Lett. Math. Phys. 45 113–9) presented classification results on three-dimensional tetrahedron maps obtained from the local Yang–Baxter equation for a certain class of matrix-functions in the situation when the equation possesses a unique solution which determines a tetrahedron map. In this paper, using correspondences arising from the local Yang–Baxter equation for some simple matrix-functions, we show that there are (non-unique) solutions to the local Yang–Baxter equation which define tetrahedron maps that do not belong to the Sergeev list; this paves the way for a new, wider classification of tetrahedron maps. We present invariants for the derived tetrahedron maps and prove Liouville integrability for some of them. Furthermore, using the approach of solving correspondences arising from the local Yang–Baxter equation, we obtain several new birational tetrahedron maps with Lax representations and invariants, including maps on arbitrary groups, a nine-dimensional map associated with a Darboux transformation for the derivative nonlinear Schrödinger equation, and a nine-dimensional generalisation of the three-dimensional Hirota map.

ЭКВИВАЛЕНТНОСТЬ УРАВНЕНИЯ ХАНТЕРА-САКСОНА И ОБОБЩЕННОГО УРАВНЕНИЯ ФЕРРОМАГНЕТИКА ГЕЙЗЕНБЕРГА

Integrable systems play an important role in modern mathematics, theoretical and mathematical physics. The display of integrable equations with exact solutions and some special solutions can provide important guarantees for the analysis of its various properties. The Hunter-Saxton equation belongs to the family of integrable systems. The extensive and interesting mathematical theory, underlying the Hunter-Saxton equation, creates active mathematical and physical research. The Hunter-Saxton equation (HSE) is a high-frequency limit of the famous Camassa-Holm equation. The physical interpretation of HSE is the propagation of weakly nonlinear orientation waves in a massive nematic liquid crystal director field. In this paper, we propose a matrix form of the Lax representation for HSE in 𝑠𝑢ሺ𝑛 ൅ 1ሻ/𝑠ሺ𝑢ሺ1ሻ ⊕ 𝑢ሺ𝑛ሻሻ - symmetric space for the case 𝑛 ൌ 2. Lax pairs, introduced in 1968 by Peter Lax, are a tool for finding conserved quantities of integrable evolutionary differential equations. The Lax representation expands the possibilities of the equation we are considering. For example, in this paper, we will use the matrix Lax representation for the HSE to establish the gauge equivalence of this equation with the generalized Heisenberg ferromagnet equation (GHFE). The famous Heisenberg Ferromagnet Equation (HFE) is one of the classical equations integrable through the inverse scattering transform. In this paper, we will consider its generalization. Andalso the connection between the decisions of the HSE and the GHFE will be presented.

Matrix Spectral Problems and Integrability Aspects of the Błaszak-Marciniak Lattice Equations

A method to derive the matrix spectral problems of the Blaszak–Marciniak lattice equations is proposed, and the matrix Lax representations of all the three-field and four-field Blaszak-Marciniak lattice equations are given explicitly. The integrability aspects of a three-field Blaszak–Marciniak lattice equation is studied as an example. To be specific, an integrable lattice hierarchy is constructed based on the matrix spectral problem of this lattice equation, the N-fold Darboux transformation and exact solutions are derived, the solitary wave structures and interaction behaviours of these exact solutions are displayed graphically, and finally the infinitely many conservation laws are listed in a standard way

Integrability Analysis of a Two-Component Burgers-Type Hierarchy

The Lax integrability of a two-component polynomial Burgers-type dynamical system is analyzed by using a differential-algebraic approach. Its linear adjoint matrix Lax representation is constructed. A related recursive operator and an infinite hierarchy of nonlinear Lax integrable dynamical systems of the Burgers–Korteweg–de-Vries type are obtained by the gradient-holonomic technique. The corresponding Lax representations are presented.

Dark Equations and Their Light Integrability

A relatively new approach to analyzing integrability, based upon differential-algebraic and symplectic techniques, is applied to some “dark equations ”of the type introduced by Boris Kupershmidt. These dark equations have unusual properties and are not particularly well-understood. In particular, dark three-component polynomial Burgers type systems are studied in detail. Their matrix Lax representations are constructed, and the related symmetry recursion operators and infinite hierarchies of integrable nonlinear dynamical systems along with their Lax representations are derived. New linear Lax spectral problems for dark integrable countable hierarchies of dynamical systems are proposed and some special cases are considered as a means of indicating that the approach presented is applicable to a far wider class of dark equations than analyzed here.

Integrability of and differential–algebraic structures for spatially 1D hydrodynamical systems of Riemann type

A differential–algebraic approach to studying the Lax integrability of a generalized Riemann type hydrodynamic hierarchy is revisited and a new Lax representation is constructed. The related bi-Hamiltonian integrability and compatible Poissonian structures of this hierarchy are also investigated using gradient-holonomic and geometric methods.The complete integrability of a new generalized Riemann hydrodynamic system is studied via a novel combination of symplectic and differential–algebraic tools. A compatible pair of polynomial Poissonian structures, a Lax representation and a related infinite hierarchy of conservation laws are obtained.In addition, the differential–algebraic approach is used to prove the complete Lax integrability of the generalized Ostrovsky–Vakhnenko and a new Burgers type system, and special cases are studied using symplectic and gradient-holonomic tools. Compatible pairs of polynomial Poissonian structures, matrix Lax representations and infinite hierarchies of conservation laws are derived.

Matrix generalizations of integrable systems with Lax integro-differential representations

We present (2+1)-dimensional generalizations of the k-constrained Kadomtsev-Petviashvili (k-cKP) hierarchy and corresponding matrix Lax representations that consist of two integro-differential operators. Additional reductions imposed on the Lax pairs lead to matrix generalizations of Davey-Stewartson systems (DS-I,DS-II,DS-III) and (2+1)-dimensional extensions of the modified Korteweg-de Vries and the Nizhnik equation. We also present an integro-differential Lax pair for a matrix version of a (2+1)-dimensional extension of the Chen-Lee-Liu equation.

The matrix Lax representation of the generalized Riemann equations and its conservation laws

It is shown that the generalized Riemann equation is equivalent with the multicomponent generalization of the Hunter–Saxton equation. New matrix and scalar Lax representation are presented for this generalization. New class of the conserved densities, which depends explicitly on the time are obtained directly from the Lax operator. The algorithm, which allows us to generate a big class of the non-polynomial conservation laws of the generalized Riemann equation is presented. Due to this new series of conservation laws of the Hunter–Saxton equation is obtained.

Hamiltonian System of New Nonlinear Lattice Equations

A 3-dimensional Lie algebra sμ(3) is obtained with the help of the known Lie algebra. Based on the sμ(3), a new discrete 3 × 3 matrix spectral problem with three potentials is constructed. In virtue of discrete zero curvature equations, a new matrix Lax representation for the hierarchy of the discrete lattice soliton equations is acquired. It is shown that the hierarchy possesses a Hamiltonian operator and a hereditary recursion operator, which implies that there exist infinitely many common commuting symmetries and infinitely many common commuting conserved functionals.

Lax-integrable Laberge–Mathieu hierarchy of supersymmetric nonlinear dynamical systems and its finite-dimensional reduction of Neumann type

A compatibly bi-Hamiltonian Laberge–Mathieu hierarchy of supersymmetric nonlinear dynamical systems is obtained by using a relation for the Casimir functionals of the central extension of a Lie algebra of superconformal even vector fields of two anticommuting variables. Its matrix Lax representation is determined by using the property of the gradient of the supertrace of the monodromy supermatrix for the corresponding matrix spectral problem. For a supersymmetric Laberge–Mathieu hierarchy, we develop a method for reduction to a nonlocal finite-dimensional invariant subspace of the Neumann type. We prove the existence of a canonical even supersymplectic structure on this subspace and the Lax–Liouville integrability of the reduced commuting vector fields generated by the hierarchy.

Infinitely many conservation laws for two integrable lattice hierarchies associated with a new discrete Schrödinger spectral problem

In this Letter, by means of considering matrix form of a new Schrödinger discrete spectral operator equation, and constructing opportune time evolution equations, and using discrete zero curvature representation, two discrete integrable lattice hierarchies proposed by Boiti et al. [J. Phys. A: Math. Gen. 36 (2003) 139] are re-derived. From the matrix Lax representations, we demonstrate the existence of infinitely many conservation laws for the two lattice hierarchies and give the corresponding conserved densities and the associated fluxes by means of formulae. Thus their integrability is further confirmed. Specially we obtain the infinitely many conservation laws for a new discrete version of the KdV equation. A connection between the conservation laws of the discrete KdV equation and the ones of the KdV equation is discussed by two examples.

New Matrix Lax Representation for a Blaszak–Marciniak Four-Field Lattice Hierarchy and Its Infinitely Many Conservation Laws

In this article, by means of considering a 4×4 discrete isospectral problem, and constructing a proper continuous time evolution equation, and using discrete zero curvature equation, a Blaszak–Marciniak four-field lattice hierarchy is re-derived. Thus a new matrix Lax representation for the hierarchy is obtained. From the new matrix Lax representation, we demonstrate the existence of infinitely many conservation laws for the lattice hierarchy and give the corresponding conserved densities and the associated fluxes formulaically. Thus its integrability is further confirmed.

Infinitely many conservation laws of two Blaszak–Marciniak three-field lattice hierarchies

In this Letter, by means of the matrix Lax representations for two Blaszak–Marciniak three-field lattice hierarchies, we demonstrate the existence of infinitely many conservation laws for the two lattice hierarchies and give the corresponding conserved density and the associated flux formulaically.

Infinitely many conservation laws for the Blaszak–Marciniak four-field integrable lattice hierarchy

In this Letter, by means of new matrix Lax representation for the Blaszak–Marciniak four-field integrable lattice hierarchy, we demonstrate the existence of infinitely many conservation laws for the lattice hierarchy and give the corresponding conserved density and the associated flux formulaically. So, its integrability is further confirmed.

Lax Pairs for the Deformed Kowalevski and Goryachev–Chaplygin Tops

We consider a quadratic deformation of the Kowalevski top. This deformation includes a new integrable case for the Kirchhoff equations recently found by one of the authors as a degeneration. A 5×5 matrix Lax pair for the deformed Kowalevski top is proposed. We also find similar deformations of the two-field Kowalevski gyrostat and the so(p,q) Kowalevski top. All our Lax pairs are deformations of the corresponding Lax representations found by Reyman and Semenov-Tian-Shansky. A similar deformation of the Goryachev–Chaplygin top and its 3×3 matrix Lax representation is also constructed.

Integrable multiwave interaction systems of ODEs

We prove complete integrability of a system of ODEs, the so called multiple 3-wave interaction equations, which are only loosely related to the 3-wave PDEs. We introduce an alternative nondegenerate matrix Lax representation which yields all the integrals of motion directly. A new general class of multiwave interaction ODEs is constructed and complete integrability proved for a nontrivial example of these.