Riemann-Hilbert Research Articles

Analytical and numerical studies for integrable and non-integrable fractional discrete modified Korteweg-de Vries hierarchies.

Under investigation in this paper is the integrable and non-integrable fractional discrete modified Korteweg-de Vries hierarchies. The linear dispersion relations, completeness relations, inverse scattering transform, and fractional soliton solutions of the integrable fractional discrete modified Korteweg-de Vries hierarchy will be explored. The inverse scattering problem will be solved accurately by constructing Gel'fand-Levitan-Marchenko equations and Riemann-Hilbert problem. The peak velocity of fractional soliton solutions will be analyzed. Numerical solutions of the non-integrable fractional averaged discrete modified Korteweg-de Vries equation, which has a simpler form than the integrable one, will be obtained by a split-step Fourier scheme.

N-soliton solutions and asymptotic analysis for the massive Thirring model in the laboratory coordinates via the Riemann-Hilbert approach

Abstract In this paper, the N-soliton solutions for massive Thirring model (MTM) in the laboratory coordinates are analyzed via the Riemann-Hilbert approach. The direct scattering including the analyticity, symmetries and asymptotic behaviors of the Jost solutions as |λ| → ∞ and λ → 0 are given. Considering that the scattering coefficients have simple zeros, the matrix Riemann-Hilbert problem, reconstruction formulas and corresponding trace formulas are also derived. Further, the N-soliton solutions in the reflectionless case are obtained explicitly in the form of determinants. The propagation characteristics of one-soliton solutions and interaction properties of two-soliton solutions are discussed. In particular, the asymptotic expressions of two-soliton solutions as |t| → ∞ are obtained, which show that the velocities and amplitudes of the asymptotic solitons don’t change before and after interaction except the position shifts. In addition, three types of bounded states for two-soliton solutions are presented with certain parametric conditions.

The Bound‐State Soliton Solution of the Generalized Inhomogeneous Hirota Equation

ABSTRACTWe develop the Riemann‐Hilbert (RH) method for the generalized inhomogeneous Hirota equation with zero boundary condition. The RH problem is related to two kinds of scattering data: simple poles and one ‐order pole. Here, we consider that when the scattering data have one or more higher order poles, the formulas of bound‐state (BS) solitons and multiple BS solitons are obtained, and the interaction between solitons and BS solitons are shown.

Hierarchical structure and N-soliton solutions of the generalized Gerdjikov–Ivanov equation via Riemann–Hilbert problem

Hierarchical structure and N-soliton solutions of the generalized Gerdjikov–Ivanov equation via Riemann–Hilbert problem

Interactions and asymptotic analysis of N-soliton solutions for the n-component generalized higher-order Sasa-Satsuma equations.

In this paper, we systematically study the N-solitons and asymptotic analysis of the integrable n-component third-fifth-order Sasa-Satsuma equations. We conduct the spectral analysis on the (n+2)-order matrix Lax pair to formulate a Riemann-Hilbert (RH) problem, which is used to generate the N-soliton solutions via the determinants. Moreover, we visually represent the interaction dynamics of multi-soliton solutions and analyze their asymptotic behaviors. Finally, we present the higher-order N-soliton solutions by dealing with the RH problem with higher-order zeros. These results will be useful to further analyze the multi-soliton structures and design the related physical experiments.

Algebro-geometric integration of the Hirota equation and the Riemann–Hilbert problem

Algebro-geometric integration of the Hirota equation and the Riemann–Hilbert problem

Soliton resolution for the generalized complex short pulse equation with the weighted Sobolev initial data

In this work, the Cauchy problem for the generalized complex short pulse equation with initial conditions in the weighted Sobolev space H(R) is studied by using the Riemann-Hilbert method and the ∂‾-steepest descent method. Based on the spectral analysis of the Lax pair, the solution of the Cauchy problem can be expressed as solution of a Riemann-Hilbert problem, which is transformed into a solvable model after a series of deformations. Finally, the long-time asymptotics and soliton resolution of the generalized complex short pulse equation in the soliton region are obtained by resorting to the ∂‾-steepest descent method. The results also indicate that the N-soliton solutions of the generalized complex short pulse equation are asymptotically stable.

On the Riemann–Hilbert problem for the reverse space-time nonlocal Hirota equation with step-like initial data

In this paper, we use the Riemann–Hilbert (RH) method to investigate the Cauchy problem of the reverse space-time nonlocal Hirota equation with step-like initial data: q(z, 0) = o(1) as z → −∞ and q(z, 0) = δ + o(1) as z → ∞, where δ is an arbitrary positive constant. We show that the solution of the Cauchy problem can be determined by the solution of the corresponding matrix RH problem established on the plane of complex spectral parameter λ. As an example, we construct an exact solution of the reverse space-time nonlocal Hirota equation in a special case via this RH problem.

On Riemann–Hilbert problem and multiple high-order pole solutions to the cubic Camassa–Holm equation

In this work, the Riemann–Hilbert (RH) problem is employed to study the multiple high-order pole solutions of the cubic Camassa–Holm (cCH) equation with the term characterizing the effect of linear dispersion under zero boundary conditions and nonzero boundary conditions. Under the reflectionless situation, we generalize the residue theorem and obtain the multiple high-order pole solutions of cCH equation by solving an algebraic system. During the process of establishing the solution of RH problem, to simplify the calculations involving the implicitly expressed of variables (x, t) in the solution, we introduce a new scale (y, t) to ensure the solution of RH problem is explicitly expressed with respect to it. Finally, the exact solutions are obtained for cases involving one high-order pole and N high-order poles.

Inverse scattering transform for the defocusing–defocusing coupled Hirota equations with non-zero boundary conditions: Multiple double-pole solutions

The inverse scattering transform for the defocusing–defocusing coupled Hirota equations with non-zero boundary conditions at infinity is thoroughly discussed. We delve into the analytical properties of the Jost eigenfunctions and scrutinize the characteristics of the scattering coefficients. To enhance our investigation of the fundamental eigenfunctions, we have derived additional auxiliary eigenfunctions with the help of the adjoint problem. Two symmetry conditions are studied to constrain the behavior of the eigenfunctions and scattering coefficients. Utilizing these symmetries, we precisely delineate the discrete spectrum and establish the associated symmetries of the scattering data. By framing the inverse problem within the context of the Riemann–Hilbert problem, we develop suitable jump conditions to express the eigenfunctions. Consequently, we have not only derived the pure soliton solutions from the defocusing–defocusing coupled Hirota equations but also provided the multiple double-pole solutions for the first time.

Mixed single, double, and triple poles solutions for the space-time shifted nonlocal DNLS equation with nonzero boundary conditions via Riemann–Hilbert approach

In this paper, we investigate the space-time shifted nonlocal derivative nonlinear Schrödinger (DNLS) equation under nonzero boundary conditions using the Riemann–Hilbert (RH) approach for the first time. To begin with, in the direct scattering problem, we analyze the analyticity, symmetries, and asymptotic behaviors of the Jost eigenfunctions and scattering matrix functions. Subsequently, we examine the coexistence of N-single, N-double, and N-triple poles in the inverse scattering problem. The corresponding residue conditions, trace formulae, θ condition, and symmetry relations of the norming constants are obtained. Moreover, we derive the exact expression for the mixed single, double, and triple poles solutions with the reflectionless potentials by solving the relevant RH problem associated with the space-time shifted nonlocal DNLS equation. Furthermore, to further explore the remarkable characteristics of soliton solutions, we graphically illustrate the dynamic behaviors of several representative solutions, such as three-soliton, two-breather, and soliton-breather solutions. Finally, we analyze the effects of shift parameters through graphical simulations.

Globality of the DPW construction for Smyth potentials in the case of SU1,1

We construct harmonic maps into SU1,1/U1 starting from Smyth potentials ξ, by the DPW method. In this method, harmonic maps are obtained from the Iwasawa factorization of a solution L of L−1dL=ξ. However, the Iwasawa factorization in the case of a noncompact group is not always global. We show that L can be expressed in terms of Bessel functions and from the asymptotic expansion of Bessel functions we solve a Riemann-Hilbert problem to give a global Iwasawa factorization. In this way we give a more direct proof of the globality of our solution than in the work of Dorfmeister-Guest-Rossman [5], while avoiding the general isomonodromy theory used by Guest-Its-Lin [11], [12].

Riemann-hilbert problem and physics-informed neural networks method for the nonlocal Sasa-Satsuma equation

Riemann-hilbert problem and physics-informed neural networks method for the nonlocal Sasa-Satsuma equation

Soliton resolution for the Ostrovsky–Vakhnenko equation

We consider the Cauchy problem of the Ostrovsky–Vakhnenko (OV) equation expressed in the new variables (y,τ)q3−q(logq)yτ−1=0 with Schwartz initial data q0(y)>0 which supports smooth and single-valued solitons. It is shown that the solution to the Cauchy problem for the OV equation can be characterized by a 3 × 3 matrix Riemann–Hilbert (RH) problem. Furthermore, by employing the ∂̄-steepest descent method to deform the RH problem into solvable models, we derive the soliton resolution for the OV equation across two space–time regions: y/τ>0 and y/τ<0. This result also implies that the N-soliton solutions of the OV equation in variables (y,τ) are asymptotically stable.

Riemann–Hilbert problem and soliton solutions with their asymptotic analysis for the focusing nonlocal Hirota equation with step-like initial data

The Riemann–Hilbert (RH) problem is developed to study the Cauchy problem of focusing space symmetric nonlocal Hirota (fsNH) equation with step-like initial data. Firstly, we analyze the characteristics of the Jost solutions and spectral functions, including analyticity, symmetry, and asymptotics as k→∞ and k→0. Then, we discuss three cases of zero point distributions of the spectral functions, and derive the expressions of spectral functions by constructing the special analytic functions and establishing the scalar RH problem. Furthermore, the constraint relations among the zeros of the spectral functions are analyzed. Next, the corresponding RH problem equipped with residual and singularity conditions is successfully derived. These singularity conditions are then transformed into a general residue condition, which reduces the problem of solving the RH problem with reflection-less potential to solving a system of algebraic equations. Ultimately, the corresponding soliton solutions of the fsNH equation and their asymptotics are derived from the solution of the RH problem.

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 $\{(x,t)|-\infty &amp;lt;x&amp;lt;+\infty , t&amp;gt;0\}$ can be divided into three different asymptotic regions based on the phase function and $\xi =x/t$. The first one is the oscillatory region $\xi &amp;lt;\frac{\alpha ^{2}}{3\beta }$, 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 $\xi&amp;gt;\frac{\alpha ^{2}}{3\beta }$, 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.

Nonlocal combined nonlinear Schrödinger–Gerdjikov–Ivanov model: Integrability, Riemann–Hilbert problem with simple and double poles, Cauchy problem with step-like initial data

In this paper, we propose three new types of the integrable nonlocal combined nonlinear Schrödinger–Gerdjikov–Ivanov (NLS-GI) models. By the Riemann–Hilbert approach, we discuss the Cauchy problem of the reverse-space-time nonlocal combined NLS-GI model with step-like initial data: u(z, 0) = o(1) for z → −∞ and u(z, 0) = A + o(1) for z → +∞, where A is an arbitrary positive constant. First of all, we give an integrable nonlocal combined NLS-GI model and its Lax pair. Then, we consider the analytical and asymptotic behaviors, symmetries, and scattering matrix of the Jost solutions. Finally, we discuss the Cauchy problem for the nonlocal combined NLS-GI model with step-like initial data.

Solitons moving on background waves of the focusing nonlinear Schrödinger equation with step-like initial condition

This work concerns the long-time asymptotic behaviors of the focusing nonlinear Schrödinger equation with step-like initial condition in present of discrete spectrum. The exact step initial-value problem with non-vanishing boundary on one side has been solved, while the step-like initial-value problem with solitons emerging remains open. We study this problem and explore the solitons moving on background waves of the focusing nonlinear Schrödinger equation by classifying all possible locations of discrete spectrum associated with the spectral functions. It is shown that there are five kinds of zones for the discrete spectrum in complex plane, which are called dumbing zone, trapping zone, trapping/waking zone, transmitting/waking zone and transmitting zone, respectively. By means of Deift–Zhou nonlinear steepest-descent method for Riemann–Hilbert problems, the long-time asymptotics of the solution along with the locations of the solitons for each case are formulated. Numerical simulations match very well with the theoretical analysis.

Uniqueness criteria for inverse scattering problem in terms of transmission matrix in boundary condition for a first order system of ordinary differential equations

The inverse scattering problem (ISP) involves the recovery of the matrix coefficient of a first-order system on the half-line from its scattering matrix. Specifically, when the matrix coefficient exhibits a triangular structure, the system possesses a Volterra-type integral transformation operator at infinity. This transformation operator facilitates the determination of the scattering matrix on the half-line through matrix Riemann-Hilbert factorization. Solving the ISP on the half-line entails reducing it to an ISP on the whole line for the considered system. This reduction involves extending the coefficients to the whole line by zero. The uniqueness criteria in terms of transmission matrix in boundary condition for the ISP is also established.

On the two nonzero boundary problems of the AB system with multiple poles

The Riemann–Hilbert method is used to study AB systems with two nonzero boundary conditions. In order to reconstruct the potentials, spectral analysis of Lax pair and the corresponding Riemann–Hilbert problem are discussed. For these two nonzero boundary problems, we consider the cases where the sectional analytic functions have double and triple poles, respectively. The derived scattering data has multiple zeros or pairs of multiple zeros, from which the relationship between the eigenfunctions at the corresponding poles can be obtained. By solving algebraic equations, we can derive the formulas of N-order solutions in different cases. As an application, we obtain different types of solitons and breathers in the reflectionless case and the dynamic analysis of these solutions is revealed.