Matrix Riemann-Hilbert Problem Research Articles

On long-time asymptotic behavior and Painlevé asymptotic to the matrix Hirota equation

Abstract The nonlinear descent method is extended to study the long-time asymptotic behavior of the matrix Hirota equation with $4\times 4$ Lax pair in Schwartz space. The implementation of spectral analysis successfully transforms the Cauchy problem of the matrix Hirota equation into the corresponding high-order Riemann–Hilbert (RH) with $4\times 4$ jump matrix, and further analyses the established oscillation RH problem to study the asymptotic behavior of the solution in the space-time plane. Interestingly, the space-time plane $\{(x,t)|-\infty <x<+\infty , t>0\}$ can be divided into three different asymptotic regions based on the phase function and $\xi =x/t$. The first one is the oscillatory region $\xi <\frac{\alpha ^{2}}{3\beta }$, whose leading-term can be approximated applying the Weber equation with an error of $\mathcal{O}(t^{-1}\log t)$. The second region is the Painlevé region $\xi \approx \frac{\alpha ^{2}}{3\beta }$, whose leading-term can be approximated by the coupled Painlevé II equation, which is related to a $4\times 4$ matrix RH problem with an error of $\mathcal{O}(t^{-\frac{2}{3}})$. The last region is the fast decay region $\xi>\frac{\alpha ^{2}}{3\beta }$, which solution is rapidly decreasing as $t\rightarrow \infty $. Our results provide a detailed proof for the asymptotic analysis for the solution of the matrix Hirota equation on the complete space-time plane.

The coupled Hirota equations with a 3×3$3\times 3$ Lax pair: Painlevé‐type asymptotics in transition zone

AbstractWe consider the Painlevé asymptotics for a solution of the integrable coupled Hirota equations with a Lax pair whose initial data decay rapidly at infinity. Using the Riemann–Hilbert (RH) techniques and Deift–Zhou nonlinear steepest descent arguments, in a transition zone defined by , where is a constant, it turns out that the leading‐order term to the solution can be expressed in terms of the solution of a coupled Painlevé II equations, which are associated with a matrix RH problem and appear in a variety of random matrix models.

The unified transformation approach to higher-order Gerdjikov-Ivanov model and Riemann-Hilbert problem

In this paper, a higher-order Gerdjikov-Ivanov (HOGI) model with cubic dispersion and quintic nonlinearity terms will be researched, which is a 3-order flow model of the well known GI system. The initial-boundary value problems of the HOGI model on the half-line will be given by the unified transformation approach. The solutions of the HOGI model will be expressed by forms of the solutions of a matrix Riemann-Hilbert problem (RHP). The relevant jump matrices of the RHP will be explicitly found by the two scattering matrices s(ϵ) and S(ϵ). Finally, the so-called global relation will be given.

Riemann–Hilbert problem for a (3+1)-dimensional nonlinear evolution equation

This paper concentrates on a (3+1)-dimensional nonlinear evolution equation. By introducing a transformation, the (3+1)-dimensional nonlinear evolution equation is decomposed into three integrable (1+1)-dimensional models. On the basis of a quartet Lax pair, we build the associated matrix Riemann–Hilbert problem. As a consequence, solving the obtained matrix Riemann–Hilbert problem with the identity jump matrix, corresponding to the reflectionless, the soliton solution to the (3+1)-dimensional nonlinear evolution equation is acquired. Specially, the one-soliton solutions are worked out and analyzed graphically.

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.

Multi-pole solitons and breathers for a nonlocal Lakshmanan-Porsezian-Daniel equation with non-zero boundary conditions

Utilizing the Riemann-Hilbert approach, we study the inverse scattering transformation, as well as multi-pole solitons and breathers, for a nonlocal Lakshmanan-Porsezian-Daniel equation with non-zero boundary conditions at infinity. Beginning with the Lax pair, we introduce the uniformization variable to simplify both the direct and inverse problems on the two-sheeted Riemann surface. In the direct scattering problem, we systematically demonstrate the analyticity, asymptotic behaviors and symmetries of the Jost functions and scattering matrix. By solving the corresponding matrix Riemann-Hilbert problem, we work out the multi-pole solutions expressed as determinants for the reflectionless potential. Based on the parameter modulation, the dynamical properties of the simple-, double- and triple-pole solutions are investigated. In the defocusing cases, we show abundant simple-pole solitons including dark solitons, anti-dark-dark solitons, double-hump solitons, as well as double- and triple-pole solitons. In addition, the asymptotic expressions for the double-pole soliton solutions are presented. In the focusing cases, we illustrate the propagations of simple-pole, double-pole, and triple-pole breathers. Furthermore, the multi-pole breather solutions can be reduced to the bright soliton solutions for the focusing nonlocal Lakshmanan-Porsezian-Daniel equation.

Robust inverse scattering analysis of discrete high-order nonlinear Schrödinger equation

Abstract In this study, we present the rigorous theory of the robust inverse scattering method for the discrete high-order nonlinear Schrödinger (HNLS) equation with a nonzero boundary condition (NZBC). Using the direct scattering problem, we deduce the analyticity, symmetries, and asymptotic behaviors of the Jost solutions and scattering matrix. We also formulate the inverse scattering problem using the matrix Riemann–Hilbert problem (RHP). Furthermore, utilizing the loop group theory, we construct the multi-fold Darboux transformation (DT) within the framework of the robust inverse scattering transform. Additionally, we develop the corresponding Bäcklund transformation (BT) to obtain the multi-fold lattice soliton solutions. To derive the high-order rational solutions, we further construct the high-order DT. Finally, we theoretically and graphically analyze these solutions, which exhibit lattice breather waves, W-shape lattice solitons, high-order lattice rogue waves (RW), and their interactions.

Inverse scattering and soliton dynamics for the mixed Chen–Lee–Liu derivative nonlinear Schrödinger equation

Under investigation in this letter is a mixed Chen–Lee–Liu derivative nonlinear Schrödinger equation which can be considered as the simplest model to approximate the dynamics of weakly nonlinear and dispersive waves, taking into account the self-steepening effect. The inverse scattering transform under the zero boundary conditions and analytical scattering coefficients with an arbitrary number of simple and double zeros is detailedly discussed. Particularly, the inverse problem is solved by the study of a matrix Riemann–Hilbert problem. As a consequence, we present the general solution for the potential, and explicit expression for the reflectionless potential.

Initial-Boundary Value Problem for the Maxwell-Bloch Equations with an Arbitrary Inhomogeneous Broadening and Periodic Boundary Function

The initial-boundary value problem (IBVP) for the Maxwell-Bloch equations with an arbitrary inhomogeneous broadening and periodic boundary condition is studied. This IBVP describes the propagation of an electromagnetic wave generated by periodic pumping in a resonant medium with distributed two-level atoms. We extended the inverse scattering transform method in the form of the matrix Riemann-Hilbert problem for solving the considered IBVP. Using the system of Ablowitz-Kaup-Newell-Segur equations equivalent to the system of the Maxwell-Bloch (MB) equations, we construct the associated matrix Riemann-Hilbert (RH) problem. Theorems on the existence, uniqueness and smoothness properties of a solution of the constructed RH problem are proved, and it is shown that a solution of the considered IBVP is uniquely defined by the solution of the associated RH problem. It is proved that the RH problem provides the causality principle. The representation of a solution of the MB equations in terms of a solution of the associated RH problem are given. The significance of this method also lies in the fact that, having studied the asymptotic behavior of the constructed RH problem and equivalent ones, we can obtain formulas for the asymptotics of a solution of the corresponding IBVP for the MB equations.

Riemann–Hilbert approach to the focusing and defocusing nonlocal derivative nonlinear Schrödinger equation with step-like initial data

In this paper, we consider the Cauchy problem for the integrable nonlocal derivative nonlinear Schrödinger (DNLS) equation with step-like initial data. The main aim is to obtain a solution for the prescribed initial conditions. To do this, we adapt the Riemann–Hilbert (RH) approach to discuss of the step-like initial value problem associated with the nonlocal DNLS equation. The main idea is the representation of the solution of the Cauchy problem in terms of the solution of an associated 2 × 2 matrix RH problem and obtain the solution of the Cauchy problem of the nonlocal DNLS equation.

The complex MKDV equation with step-like initial data: Large time asymptotic analysis

In this paper, we study large-time asymptotics for the complex modified Korteveg–de Vries equation with step-like initial data. It is shown that the step-like initial problem can be described by a matrix Riemann–Hilbert problem. Further we apply the steepest descent method to obtain different large-time asymptotics in the Zakharov–Manakov region, a plane wave region, and a slow decay region.

Double and triple-pole solutions for the third-order flow equation of the Kaup-Newell system with zero/nonzero boundary conditions

In this work, the double and triple-pole solutions for the third-order flow equation of Kaup-Newell system (TOFKN) with zero boundary conditions (ZBCs) and non-zero boundary conditions (NZBCs) are investigated by means of the Riemann-Hilbert (RH) approach stemming from the inverse scattering transformation. Starting from spectral problem of the TOFKN, the analyticity, symmetries, asymptotic behavior of the Jost function and scattering matrix, the matrix RH problem with ZBCs and NZBCs are constructed. Then the obtained RH problem with ZBCs and NZBCs can be solved in the case of scattering coefficients with double or triple zeros, and the reconstruction formula of potential, trace formula as well as theta condition are also derived correspondingly. Specifically, the general formulas of N-double and N-triple poles solutions with ZBCs and NZBCs are derived systematically by means of determinants. The vivid plots and dynamics analyses for double and triple-pole soliton solutions with the ZBCs as well as double and triple-pole interaction solutions with the NZBCs are exhibited in details. Compared with the most classical second-order flow Kaup-Newell system, we find the third-order dispersion and quintic nonlinear term of the Kaup-Newell system change the trajectory and velocity of solutions. Furthermore, the asymptotic states of the 1-double poles soliton solution and the 1-triple poles soliton solution are analyzed when t tends to infinity.

A coupled complex mKdV equation and its N-soliton solutions via the Riemann–Hilbert approach

This paper concerns the initial value problem of a coupled complex mKdV (CCMKDV) equations ut+uxxx+6(|u|2+|v|2)ux+6u(|v|2)x=0,vt+vxxx+6(|u|2+|v|2)vx+6v(|u|2)x=0,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\begin{aligned} &u_{t}+u_{xxx}+6 \\bigl( \\vert u \\vert ^{2}+ \\vert v \\vert ^{2} \\bigr)u_{x}+6u\\bigl( \\vert v \\vert ^{2} \\bigr)_{x}=0, \\\\ &v_{t}+v_{xxx}+6\\bigl( \\vert u \\vert ^{2}+ \\vert v \\vert ^{2}\\bigr)v_{x}+6v \\bigl( \\vert u \\vert ^{2}\\bigr)_{x}=0, \\end{aligned} $$\\end{document} proposed by Yang (Nonlinear Waves in Integrable and Nonintegrable Systems, 2010), which is associated with a 4 times 4 scattering problem. Based on matrix spectral analysis, a fourth-order matrix Riemann–Hilbert problem is formulated. By solving a specific nonregular Riemann–Hilbert problem with zeros, we present the N-soliton solutions for the CCMKDV system. Moreover, the single-soliton solutions are displayed graphically.

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.

The long-time asymptotics of the derivative nonlinear Schr[formula omitted]dinger equation with step-like initial value

Consideration in this present paper is the long-time asymptotics of solutions to the derivative nonlinear Schrödinger (DNLS) equation with the step-like initial value q(x,0)=q0(x)=A1eiϕe2iBx,x<0,A2e−2iBx,x>0,by Deift–Zhou method. The step-like initial problem is described by a matrix Riemann–Hilbert problem. A crucial ingredient used in this paper is to introduce the g-function mechanism for solving the problem of the entries of the jump matrix growing exponentially as t→∞. It is shown that the leading order term of the long-time asymptotics solution of the DNLS equation is expressed by the Theta function Θ about the Riemann-surface of genus 3 and the subleading order term expressed by parabolic cylinder and Airy functions.

Inverse scattering transform for the two‐component Gerdjikov–Ivanov equation with nonzero boundary conditions: Dark‐dark solitons

AbstractThe two‐component Gerdjikov–Ivanov equation with nonzero boundary conditions is studied by the inverse scattering transform. A fundamental set of analytic eigenfunctions is obtained with the aid of the associated adjoint problem. Three symmetry conditions are discussed to curb the scattering data. The behavior of the Jost functions and the scattering matrix at the branch points is discussed. The inverse scattering problem is formulated by a matrix Riemann–Hilbert problem. The trace formula in terms of the scattering data and the so‐called asymptotic phase difference for the potential are obtained. The solitons classification is described in detail. When the discrete eigenvalues lie on the circle, the dark‐dark soliton is obtained for the first time in this work. And the discrete eigenvalues off the circle generate the dark‐bright, bright‐bright, breather‐breather, M(‐type)‐W(‐type) solitons, and their interactions.

Riemann-Hilbert problem for the fifth-order modified Korteweg–de Vries equation with the prescribed initial and boundary values

In this paper, we investigate the fifth-order modified Korteweg–de Vries (mKdV) equation on the half-line via the Fokas unified transformation approach. We show that the solution u(x, t) of the fifth-order mKdV equation can be represented by the solution of the matrix Riemann-Hilbert problem constructed on the plane of complex spectral parameter θ. The jump matrix L(x, t, θ) has an explicit representation dependent on x, t and it can be represented exactly by the two pairs of spectral functions y(θ), z(θ) (obtained from the initial value u 0(x)) and Y(θ), Z(θ) (obtained from the boundary conditions v 0(t), ). Furthermore, the two pairs of spectral functions y(θ), z(θ) and Y(θ), Z(θ) are not independent of each other, but are related to the compatibility condition, the so-called global relation.

Riemann–Hilbert problems and N-soliton solutions of the nonlocal reverse space-time Chen–Lee–Liu equation

In this paper, the N-soliton solutions to the nonlocal reverse space-time Chen–Lee–Liu equation have been derived. Under the nonlocal symmetry reduction to the matrix spectral problem, the nonlocal reverse space-time Chen–Lee–Liu equation can be obtained. Based on the spectral problem, the specific matrix Riemann–Hilbert problem is constructed for this nonlocal equation. Through solving this associated Riemann–Hilbert problem, the N-soliton solutions to this nonlocal equation can be obtained in the case of the jump matrix as an identity matrix.

Interactions of fractional [formula omitted]-solitons with anomalous dispersions for the integrable combined fractional higher-order mKdV hierarchy

In this paper, we investigate the anomalous dispersive relations, inverse scattering transform with a Riemann–Hilbert (RH) problem, and fractional multi-solitons of the integrable combined fractional higher-order mKdV (fhmKdV) hierarchy, including the fractional mKdV (fmKdV), fractional fifth-order mKdV (f5mKdV), fractional combined third–fifth-order mKdV (f35mKdV) equations, etc., which can be featured via completeness of squared scalar eigenfunctions of the ZS spectral problem. We construct a matrix RH problem to present three types of fractional N-solitons illustrating anomalous dispersions of the combined fhmKdV hierarchy for the reflectionless case. As some examples, we analyse the wave velocity of the fractional one-soliton such that we find that the fhmKdV equation predicts a power law relationship between the wave velocity and amplitude, and demonstrates the anomalous dispersion. Furthermore, we illustrate other interesting anomalous dispersive wave phenomena containing the elastic interactions of fractional bright and dark solitons, W-shaped soliton and dark soliton, as well as breather and dark soliton. These obtained fractional multi-solitons will be useful to understand the related nonlinear super-dispersive wave propagations in fractional nonlinear media.

The mixed nonlinear Schrödinger equation on the half-line

We analyze the initial-boundary value problem for the mixed nonlinear Schrödinger equation posed on the half-line by using the Fokas method. Assuming that a smooth solution exists, we show that the solution can be represented in term of the solution of a matrix Riemann–Hilbert problem formulated in the complex plane. The jump matrix is defined in terms of the spectral functions obtained from the initial and boundary values. We derive a certain relation, the so-called global relation that involves all initial and boundary values. Furthermore, it can be shown that the solution for the mixed nonlinear Schrödinger equation posed on the half-line exists, which can be determined by the unique solution of the associated Riemann–Hilbert problem, as long as the spectral functions satisfy the above global relation.