Matrix Spectral Problem Research Articles

A new lattice hierarchy: Hamiltonian structures, symplectic map and N-fold Darboux transformation

• A new integrable lattice hierarchy is proposed. • The associated Hamiltonian structures and symplectic map are derived. • The N -fold Douboux transformations of the first two lattice systems are given. • The elastic and inelastic interaction phenomena in the lattice systems are found. A new lattice hierarchy is constructed from a discrete matrix spectral problem. By the Tu scheme technique, the associated Hamiltonian structures and infinitely many conservation laws of this hierarchy are derived. Then a symplectic map is proposed based on the Lax pair and the adjoint Lax pair. Furthermore, the N -fold Darboux transformation and explicitly exact solutions of the first two equations in the hierarchy are investigated. Finally, the density profiles of these exact solutions are presented to illustrate the overtaking collisions of solitary waves .

Application of the Riemann–Hilbert approach to the multicomponent AKNS integrable hierarchies

A class of Riemann–Hilbert problems on the real axis is formulated for solving the multicomponent AKNS integrable hierarchies associated with a kind of bock matrix spectral problems. Through special Riemann–Hilbert problems where a jump matrix is taken to be the identity matrix, soliton solutions to all integrable equations in each hierarchy are explicitly computed. A class of specific reductions of the presented integrable hierarchies is also made, together with its reduced Lax pairs and soliton solutions.

A τ-Symmetry Algebra of the Generalized Derivative Nonlinear Schrödinger Soliton Hierarchy with an Arbitrary Parameter

A matrix spectral problem is researched with an arbitrary parameter. Through zero curvature equations, two hierarchies are constructed of isospectral and nonisospectral generalized derivative nonlinear schrödinger equations. The resulting hierarchies include the Kaup-Newell equation, the Chen-Lee-Liu equation, the Gerdjikov-Ivanov equation, the modified Korteweg-de Vries equation, the Sharma-Tasso-Olever equation and a new equation as special reductions. The integro-differential operator related to the isospectral and nonisospectral hierarchies is shown to be not only a hereditary but also a strong symmetry of the whole isospectral hierarchy. For the isospectral hierarchy, the corresponding τ -symmetries are generated from the nonisospectral hierarchy and form an infinite-dimensional symmetry algebra with the K-symmetries.

Riemann surface and Riemann theta function solutions of the discrete integrable hierarchy

A hierarchy of discrete nonlinear evolution equations associated with a discrete 3 × 3 matrix spectral problem with two potentials is proposed by means of the Lenard recursion equations and zero-curvature equation. Based on the characteristic polynomial of Lax matrix for the hierarchy, we introduce a trigonal curve and study the properties of the corresponding three-sheeted Riemann surface, especially including arithmetic genus, holomorphic differentials. Base on the essential properties of the meromorphic functions ϕ2, ϕ3 and the Baker–Akhiezer function ψ1, and their asymptotic behavior, we obtain Riemann theta function solutions of the entire discrete integrable hierarchy.

Completion of the Ablowitz-Kaup-Newell-Segur integrable coupling

Integrable couplings are associated with non-semisimple Lie algebras. In this paper, we propose a new method to generate new integrable systems through making perturbation in matrix spectral problems for integrable couplings, which is called the “completion process of integrable couplings.” As an example, the idea of construction is applied to the Ablowitz-Kaup-Newell-Segur integrable coupling. Each equation in the resulting hierarchy has a bi-Hamiltonian structure furnished by the component-trace identity.

Transfer matrix spectrum for cyclic representations of the 6-vertex reflection algebra II

This article is a direct continuation of where we begun the study of the transfer matrix spectral problem for the cyclic representations of the trigonometric 6-vertex reflection algebra associated to the Bazhanov-Stroganov Lax operator. There we addressed this problem for the case where one of the KK-matrices describing the boundary conditions is triangular. In the present article we consider the most general integrable boundary conditions, namely the most general boundary KK-matrices satisfying the reflection equation. The spectral analysis is developed by implementing the method of Separation of Variables (SoV). We first design a suitable gauge transformation that enable us to put into correspondence the spectral problem for the most general boundary conditions with another one having one boundary KK-matrix in a triangular form. In these settings the SoV resolution can be obtained along an extension of the method described in . The transfer matrix spectrum is then completely characterized in terms of the set of solutions to a discrete system of polynomial equations in a given class of functions and equivalently as the set of solutions to an analogue of Baxter’s T-Q functional equation. We further describe scalar product properties of the separate states including eigenstates of the transfer matrix.

On quantum separation of variables

We present a new approach to construct the separate variables basis leading to the full characterization of the transfer matrix spectrum of quantum integrable lattice models. The basis is generated by the repeated action of the transfer matrix itself on a generically chosen state of the Hilbert space. The fusion relations for the transfer matrix, stemming from the Yang-Baxter algebra properties, provide the necessary closure relations to define the action of the transfer matrix on such a basis in terms of elementary local shifts, leading to a separate transfer matrix spectral problem. Hence our scheme extends to the quantum case a key feature of the Liouville-Arnold classical integrability framework where the complete set of conserved charges defines both the level manifold and the flows on it leading to the construction of action-angle variables. We work in the framework of the quantum inverse scattering method. As a first example of our approach, we give the construction of such a basis for models associated with Y(gln) and argue how it extends to their trigonometric and elliptic versions. Then we show how our general scheme applies concretely to fundamental models associated with the Y(gl2) and Y(gl3) R-matrices leading to the full characterization of their spectrum. For Y(gl2) and its trigonometric deformation, a particular case of our method reproduces Sklyanin’s construction of separate variables. For Y(gl3), it gives new results, in particular, through the proper identification of the shifts acting on the separate basis. We stress that our method also leads to the full characterization of the spectrum of other known quantum integrable lattice models, including, in particular, trigonometric and elliptic spin chains, open chains with general integrable boundaries, and further higher rank cases that we will describe in forthcoming publications.

Riemann–Hilbert problems of a six-component fourth-order AKNS system and its soliton solutions

Associated with a $$4\times 4$$ matrix spectral problem, a six-component AKNS soliton hierarchy is presented, together with the first three nonlinear soliton systems. From an equivalent spectral problem, a kind of Riemann–Hilbert problems is formulated for a six-component system of fourth-order AKNS equations in the resulting AKNS hierarchy. Soliton solutions to the considered system of coupled fourth-order AKNS equations are worked out from a reduced Riemann–Hilbert problem where an identity jump matrix is taken.

Generalized Integrable Hierarchies of AKNS Type, Super Dirac Type and Super NLS–mKdV Type

Integrable hierarchies provide many important physical models. Firstly, with the help of symbolic computation software Maple, two generalized integrable hierarchies of Ablowitz–Kaup–Newell–Segur (AKNS) type are constructed from the matrix spectral problem associated with the Lie algebra s l ( 2 ) . Then, two generalized integrable hierarchies of super Dirac type and super nonlinear Schrodinger-modified Korteweg–de Vries (NLS–mKdV) type are obtained to illustrate the use of the super Lie algebra B ( 0 , 1 ) .

A Family of Integrable Different-Difference Equations, Its Hamiltonian Structure, and Darboux-Bäcklund Transformation

An integrable family of the different-difference equations is derived from a discrete matrix spectral problem by the discrete zero curvature representation. Hamiltonian structure of obtained integrable family is established. Liouville integrability for the obtained family of discrete Hamiltonian systems is proved. Based on the gauge transformation between the Lax pair, a Darboux-Bäcklund transformation of the first nonlinear different-difference equation in obtained family is deduced. Using this Darboux-Bäcklund transformation, an exact solution is presented.

Riemann–Hilbert problems and N-soliton solutions for a coupled mKdV system

A 3 × 3 matrix spectral problem is introduced and its associated AKNS integrable hierarchy with four components is generated. From this spectral problem, a kind of Riemann–Hilbert problems is formulated for a system of coupled mKdV equations in the resulting AKNS integrable hierarchy. N -soliton solutions to the coupled mKdV system are presented through a specific Riemann–Hilbert problem with an identity jump matrix.

A super Sasa–Satsuma hierarchy and bi-Hamiltonian structures

A super Sasa–Satsuma hierarchy associated with a 3 × 3 matrix spectral problem is proposed with the aid of the zero-curvature equation and Lenard recursion equations. A typical member in the hierarchy is the super Sasa–Satsuma equation. The super bi-Hamiltonian structures of the super Sasa–Satsuma hierarchy are constructed by utilizing the super trace identity. The infinite conservation laws of the super Sasa–Satsuma equation are presented by resorting to the spectral parameter expansion.

A new nonlinear wave equation: Darboux transformation and soliton solutions

In this paper, a new nonlinear wave equation associated with a 2 × 2 matrix spectral problem is proposed by means of the zero-curvature equation and the polynomial expansion of the spectral parameter. With the help of a gauge transformation between the corresponding Lax pair, a Darboux transformation of the nonlinear wave equation is obtained. As an application, by taking different seed solutions and using the Darboux transformation, one can get a variety of types of exact solutions for the nonlinear wave equation, like one-soliton solution, two-soliton solution, periodic solution, and Akhmediev breather solution.

Transfer matrix spectrum for cyclic representations of the 6-vertex reflection algebra by quantum separation of variables

In this proceeding, we recall the notion of quantum integrable systems on a lattice and then introduce the Sklyanin’s Separation of Variables method. We sum up the main results for the transfer matrix spectral problem for the cyclic representations of the trigonometric 6-vertex reflection algebra associated to the Bazanov-Stroganov Lax operator. These results apply as well to the spectral analysis of the lattice sine-Gordon model with open boundary conditions. The transfer matrix spectrum (both eigenvalues and eigenstates) is completely characterized in terms of the set of solutions to a discrete system of polynomial equations. We state an equivalent characterization as the set of solutions to a Baxter’s like T-Q functional equation, allowing us to rewrite the transfer matrix eigenstates in an algebraic Bethe ansatz form.

An extension of the coupled derivative nonlinear Schrödinger hierarchy

In this paper, a 3 × 3 matrix spectral problem with six potentials is considered. With the help of the compatibility condition, a hierarchy of new nonlinear evolution equations which can be reduced to the coupled derivative nonlinear Schrödinger (CDNLS) equations is obtained. By use of the trace identity, it is proved that all the members in this new hierarchy have generalized bi-Hamiltonian structures. Moreover, infinitely many conservation laws of this hierarchy are constructed.

Nonlinear integrable couplings of a generalized super Ablowitz‐Kaup‐Newell‐Segur hierarchy and its super bi‐Hamiltonian structures

In this paper, a new generalized 5×5 matrix spectral problem of Ablowitz‐Kaup‐Newell‐Segur type associated with the enlarged matrix Lie superalgebra is proposed, and its corresponding super soliton hierarchy is established. The super variational identities are used to furnish super Hamiltonian structures for the resulting super soliton hierarchy.

Finite-Band Solutions for the Hierarchy of Coupled Toda Lattices

Based on the characteristic polynomial of Lax matrix for the hierarchy of coupled Toda lattices associated with a \(3\times3\) discrete matrix spectral problem, we introduce a trigonal curve with two infinite points, from which we establish the associated Dubrovin-type equations. The asymptotic properties of the meromorphic function and the Baker-Akhiezer function are studied near two infinite points on the trigonal curve. Finite-band solutions of the entire hierarchy of coupled Toda lattices are obtained in terms of the Riemann theta function.

The Nonlinear Steepest Descent Method to Long-Time Asymptotics of the Coupled Nonlinear Schrödinger Equation

The Riemann–Hilbert problem for the coupled nonlinear Schrodinger equation is formulated on the basis of the corresponding $$3\times 3$$ matrix spectral problem. Using the nonlinear steepest descent method, we obtain leading-order asymptotics for the Cauchy problem of the coupled nonlinear Schrodinger equation.

An integrable lattice hierarchy based on Suris system: $${\varvec{N}}$$ N -fold Darboux transformation and conservation laws

An integrable lattice hierarchy is constructed from a discrete matrix spectral problem, in which one of the Suris systems is the first member of this hierarchy. Some related properties such as Hamiltonian structure of this lattice hierarchy are discussed. The Suris system is solved by the N-fold Darboux transformation. As a result, the multi-soliton solutions are derived and the soliton structures along with the interaction behaviors among solitons are shown graphically. Finally, the infinitely many conservation laws of the Suris system are given.

Two symmetry constraints for a generalized Dirac integrable hierarchy

By a two-by-two matrix spectral problem, a generalized Dirac integrable hierarchy is presented. A Hamiltonian structure of the obtained hierarchy is established by trace identity, and its Liouville integrability is proved. Then, through Bargmann symmetry constraint, spatial part of the Lax pairs and adjoint Lax pairs is nonlinearized as a completely integrable finite-dimensional Hamiltonian system. Next, under an implicit symmetry constraint, both spatial part and temporal parts of the Lax pairs and adjoint Lax pairs are all nonlinearized as completely integrable finite-dimensional Hamiltonian systems. Ultimately, the involutive representation of solution of the generalized Dirac integrable hierarchy is given.