Matrix Spectral Problem Research Articles

Long-time asymptotic behavior for the Hermitian symmetric space derivative nonlinear Schrödinger equation

Abstract Resorting to the spectral analysis of the 4 × 4 matrix spectral problem, we construct a 4 × 4 matrix Riemann–Hilbert problem to solve the initial value problem for the Hermitian symmetric space derivative nonlinear Schrödinger equation. The nonlinear steepest decent method is extended to study the 4 × 4 matrix Riemann–Hilbert problem, from which the various Deift–Zhou contour deformations and the motivation behind them are given. Through some proper transformations between the corresponding Riemann–Hilbert problems, the basic Riemann–Hilbert problem is reduced to a model Riemann–Hilbert problem, by which the long-time asymptotic behavior to the solution of the initial value problem for the Hermitian symmetric space derivative nonlinear Schrödinger equation is obtained with the help of the asymptotic expansion of the parabolic cylinder function and strict error estimates.

A combined Liouville integrable hierarchy associated with a fourth-order matrix spectral problem

This paper aims to propose a fourth-order matrix spectral problem involving four potentials and generate an associated Liouville integrable hierarchy via the zero curvature formulation. A bi-Hamiltonian formulation is furnished by applying the trace identity and a recursion operator is explicitly worked out, which exhibits the Liouville integrability of each model in the resulting hierarchy. Two specific examples, consisting of novel generalized combined nonlinear Schrödinger equations and modified Korteweg–de Vries equations, are given.

[formula omitted]-soliton solutions and their dynamic analysis to the generalized complex mKdV equation

A new generalized complex modified Korteweg–de Vries (mKdV) equation is studied by using Riemann-Hilbert approach. Firstly, we derive a Lax pair associated with a 3 × 3 matrix spectral problem for the generalized complex mKdV equation. Then, we can formulate the Riemann-Hilbert problem via the spectral analysis of the x-part of the Lax pair. According to the symmetry properties of the potential matrix, we find two cases of zero structures for the Riemann-Hilbert problem. By solving the particular Riemann-Hilbert problem and using the inverse scattering transformation, we obtain the unified formulas of the N-soliton solutions for the generalized complex mKdV equation. In addition, the dynamical behaviors of the single-soliton solution and the two-soliton solution are analyzed by choosing appropriate parameters.

An integrable generalization of the Fokas–Lenells equation: Darboux transformation, reduction and explicit soliton solutions

Abstract Under investigation is an integrable generalization of the Fokas–Lenells equation, which can be derived from the negative power flow of a 2 × 2 matrix spectral problem with three potentials. Based on the gauge transformation of the matrix spectral problem, one kind of Darboux transformation with multi-parameters for the three-component coupled Fokas–Lenells system is constructed. As a reduction, the N-fold Darboux transformation for the generalized Fokas–Lenells equation is obtained, from which the N-soliton solution in a compact Vandermonde-like determinant form is given. Particularly, the explicit one- and two-soliton solutions are presented and their dynamical behaviors are shown graphically.

A Generalized Hierarchy of Combined Integrable Bi-Hamiltonian Equations from a Specific Fourth-Order Matrix Spectral Problem

The aim of this paper is to analyze a specific fourth-order matrix spectral problem involving four potentials and two free nonzero parameters and construct an associated integrable hierarchy of bi-Hamiltonian equations within the zero curvature formulation. A hereditary recursion operator is explicitly computed, and the corresponding bi-Hamiltonian formulation is established by the so-called trace identity, showing the Liouville integrability of the obtained hierarchy. Two illustrative examples are novel generalized combined nonlinear Schrödinger equations and modified Korteweg–de Vries equations with four components and two adjustable parameters.

On behavior analysis of solutions for the coupled higher-order WKI equation

A coupled higher-order WKI equation is proposed and its 4 × 4 matrix Lax pair is derived. On the basis of spectral analysis of the 4 × 4 matrix spectral problems related to the coupled higher-order WKI equation, the basic Riemann–Hilbert problem is established by using the inverse scattering transformation. Resorting to the nonlinear steepest descent method, the solution of the basic Riemann–Hilbert problem is transformed to the solution of the model Riemann–Hilbert problem after several contour deformation. Finally, we explicitly obtain the long-time asymptotics of the solution to the Cauchy problem of the coupled higher-order WKI equation in the solitonless sector region with the aid of the parabolic cylinder function.

An extended Merola–Ragnisco–Tu lattice integrable hierarchy and infinite conservation laws

By means of the zero-curvature equation and Lenard recursive operators, a novel differential-difference integrable hierarchy is derived, which is related to a discrete 3 × 3 matrix spectral problem with five potentials. The first nontrivial member in this hierarchy under specific reduction is the Merola–Ragnisco–Tu lattice equation whose continuum limit is the nonlinear Schrödinger equation. The infinite conservation laws of the first two nontrivial members in the hierarchy, namely, the extended Merola–Ragnisco–Tu lattice equation and its high-order generalization are constructed with the help of two Ricatti-type equations and the spectral parameter expansions.

Long-time asymptotics for the coupled complex short-pulse equation with decaying initial data

We characterize the long-time asymptotic behavior of the solution of the initial value problem for the coupled complex short-pulse equation associated with the 4×4 matrix spectral problem. The spectral analysis of the 4×4 matrix spectral problem is very difficult because of the existence of energy-dependent potentials and the WKI type. The method we adopted is a combination of the inverse scattering transform and Deift-Zhou nonlinear steepest descent method. Starting from the Lax pair associated with the coupled complex short-pulse equation, we derive a basic Riemann-Hilbert problem by introducing some appropriate spectral function transformations, and reconstruct the potential parameterized from the solution of the basic Riemann-Hilbert problem via the asymptotic behavior of the spectral variable at k→0. We finally obtain the leading order asymptotic behavior of the solution of the coupled complex short-pulse equation through a series of Deift-Zhou contour deformations.

Novel Liouville integrable Hamiltonian models with six components and three signs

This paper aims at constructing a six-component integrable hierarchy associated with a matrix spatial spectral problem with six potentials and three signs. The zero curvature formulation and the trace identity are used to generate integrable models and their Hamiltonian structures, respectively. Two expository examples of integrable models of lower orders are six-component integrable coupled nonlinear Schrödinger (NLS) equations and modified Korteweg–de Vries (mKdV) equations. The motivation of this study is to explore typical integrable coupled NLS equations and mKdV equations, and the innovative idea and main advance is to introduce a specific matrix spectral problem involving three signs to construct integrable coupled equations.

A Liouville integrable hierarchy with four potentials and its bi-Hamiltonian structure

"We aim to construct a Liouville integrable Hamiltonian hierarchy from a specific matrix spectral problem with four potentials through the zero curvature formulation. The Liouville integrability of the resulting hierarchy is exhibited by a bi-Hamiltonian structure explored by using the trace identity. Illustrative examples of novel four-component coupled Liouville integrable nonlinear Schr¨odinger equations and modified Korteweg-de Vries equations are presented."

Four-component integrable hierarchies and their Hamiltonian structures

We aim to construct four-component integrable hierarchies from a kind of matrix spectral problems within the zero curvature formulation. The Liouville integrability of the resulting hierarchies are guaranteed through establishing Hamiltonian structures by the trace identity. Illustrative examples include novel four-component nonlinear Schrödinger type equations and modified Korteweg–de Vries type equations.

Spectral and soliton structures for the four-component Kaup–Newell type negative flow equation

Based on the spectral analysis of Lax pairs and the inverse scattering method, we construct a matrix Riemann–Hilbert problem of the four-component Kaup–Newell type negative flow equation associated with a 4 × 4 matrix spectral problem and investigate the evolution of scattering data. Then N-soliton formulas of the four-component Kaup–Newell type negative flow equation are obtained by solving the irregular Riemann–Hilbert problem in the reflectionless case. As an application, the one-soliton solutions and the two-soliton solutions of the four-component Kaup–Newell type negative flow equation are given explicitly. In addition, the interaction dynamics of the soliton solutions are analyzed and illustrated.

New integrable peakon equation and its dynamic system

A two component generalization Camassa–Holm equation with peakon is proposed, which is associated with a 2 × 2 matrix spectral problem with two potentials. As its reduction, a new integrable peakon equation is derived. Then positive flows associated with the 2 × 2 matrix spectral problem are discussed, a new hierarchy of nonlinear evolution equations is obtained. Subsequently, N-peakon of the peakon equation are constructed and the dynamical system with which the N-peakon evolve is deduced. As an example, exact single peakon and 2-peakon are given and their figures are plotted. Moreover, infinitely many conserved quantities of the peakon equation are obtained.

AKNS type reduced integrable bi-Hamiltonian hierarchies with four potentials

The aim of this paper is to derive a kind of integrable hierarchies with four potentials, which possess bi-Hamiltonian structures, from reduced AKNS matrix spectral problems. The associated recursion operators are worked out explicitly. The Lax pair formulation and the trace identity are basic tools in the analysis. Two nonlinear examples in the resulting integrable hierarchies are integrable nonlinear Schrödinger type equations and integrable modified Korteweg–de Vries type equations.

Rogue waves and breathers of the derivative Yajima-Oikawa long wave-short wave equations on theta-function backgrounds

Based on the Baker-Akhiezer functions on Riemann surfaces, a general method is proposed to construct rogue-wave solutions and breather solutions of integrable nonlinear evolution equations associated with higher-order matrix spectral problems on theta-function periodic backgrounds. The derivative Yajima-Oikawa long wave-short wave equation associated with a 3×3 matrix Lax pair is chosen as an illustrative example of our method. First, a theta-function seed solution for the derivative Yajima-Oikawa long wave-short wave equation is derived by using the Weierstrass functions. Then, the general solution of the 3×3 matrix spectral problem and its derivatives with respect to the spectral parameter are obtained with the help of the Baker-Akhiezer functions. Finally, by means of Darboux transformations, explicit exact solutions such as rogue waves and breathers on theta-function backgrounds for the derivative Yajima-Oikawa long wave-short wave equation are constructed. Taking the advantage of the Jacobi θ-functions, these new solutions are numerically calculated and illustrated. In addition, by observing the explicit solutions, it is found that two rogue waves of different shapes may appear in one solution, which is a new phenomenon different from the solutions on constant backgrounds.

Soliton hierarchies and soliton solutions of type ([formula omitted], [formula omitted] ) reduced nonlocal nonlinear Schrödinger equations of arbitrary even order

We present mixed-type reduced soliton hierarchies of nonlocal integrable nonlinear Schrödinger equations of arbitrary even order by conducting two nonlocal group reductions for the Ablowitz–Kaup–Newell–Segur matrix spectral problems. Based on specific distributions of eigenvalues and adjoint eigenvalues, we construct soliton solutions by solving the corresponding reflectionless generalized Riemann–Hilbert problems, where eigenvalues could equal adjoint eigenvalues.

Nonlocal reduced integrable mKdV-type equations from a vector integrable hierarchy

This paper aims to present two hierarchies of nonlocal reduced integrable mKdV-type equations from a vector integrable hierarchy associated with a matrix Lie algebra, not being A type. The key point is to make similarity transformations for the spectral matrix, which keep the associated zero curvature equations invariant and then there follow reduced nonlocal integrable mKdV-type equations. The success lies in determining a Laurent series solution to the corresponding reduced stationary zero curvature equation, which generates temporal matrix spectral problems in the zero curvature formation.

Higher-Order Matrix Spectral Problems and Their Integrable Hamiltonian Hierarchies

Starting from a kind of higher-order matrix spectral problems, we generate integrable Hamiltonian hierarchies through the zero-curvature formulation. To guarantee the Liouville integrability of the obtained hierarchies, the trace identity is used to establish their Hamiltonian structures. Illuminating examples of coupled nonlinear Schrödinger equations and coupled modified Korteweg–de Vries equations are worked out.

Bright-dark soliton solutions and their elastic interaction analysis for a reduced integrable spin Hirota–Maxwell–Bloch equation

Under consideration is a reduced integrable spin Hirota–Maxwell–Bloch (rsHMB) equation, which may have potential applications in describing femtosecond pulse propagation through an erbium doped fiber. With the aid of symbolic computation, we establish the N-fold Darboux transformation (DT) of rsHMB equation based on its known 2 × 2 matrix spectral problem. As an application of the resulting DT, we can get bright-dark soliton and breather solutions of rsHMB equation. Through graphical and asymptotic analysis, the soliton elastic collisions are shown and discussed graphically, and some important soliton physical features including wave velocity, amplitude and energy are analyzed. The consequences and properties presented in this paper may be helpful to comprehend the transmission of optical soliton in erbium doped fiber.

Degenerate and bound-state solitons of a novel Kundu-nonlinear Schrödinger equation based on generalized Darboux transformation

In this paper, we derive a novel Kundu-nonlinear Schrödinger equation through a 2 × 2 matrix spectral problem, which can be used to describe the phenomena in optical propagation, and the determinant expression of generalized (n,N−n)-fold Darboux transformation for the Kundu-nonlinear Schrödinger equation is given by solving algebraic equations. In addition, through the generalized (n,N−n)-fold Darboux transformation, we obtain three different types of solutions, (i) soliton solutions, (ii) degenerate soliton solutions, (iii) interaction solutions between solitons and degenerate solitons, which are divided into elastic interaction and bound-state. At the same time, the conditions for their generation are discussed. The properties of the aforementioned solutions are analyzed graphically ultimately.