Riemann-Hilbert Method Research Articles

High-order soliton solutions and their dynamics in the inhomogeneous variable coefficients Hirota equation

A series of new soliton solutions is presented for the inhomogeneous variable coefficient Hirota equation by using the Riemann–Hilbert method and transformation relationship. Firstly, through a standard dressing procedure, the N-soliton matrix associated with the simple zeros in the Riemann–Hilbert problem for the Hirota equation is constructed. Then the N-soliton matrix of the inhomogeneous variable coefficient Hirota equation can be obtained by a special relationship transformation from the N-soliton matrix of the Hirota equation. Next, using the generalized Darboux transformation, the high-order soliton solutions corresponding to the elementary high-order zeros in the Riemann–Hilbert problem for the Hirota equation can be derived. Similarly, employing the relationship transformation mentioned above can lead to the high-order soliton solutions of the inhomogeneous variable coefficient Hirota equation. In addition, the collision dynamics of Hirota and inhomogeneous variable coefficient Hirota equations are analyzed; the asymptotic behaviors for multi-solitons and long-term asymptotic estimates for the high-order one-soliton of the Hirota equation are concretely calculated. Most notably, by analyzing the dynamics of the multi-solitons and high-order solitons of the inhomogeneous variable coefficient Hirota equation, we discover numerous new waveforms such as heart-shaped periodic wave solutions, O-shaped periodic wave solutions etc. that have never been reported before, which are crucial in theory and practice.

Double-pole solutions in the modified nonlinear Schrödinger equation

This paper is concerned with the modified nonlinear Schrödinger equation with zero boundary condition or non-zero boundary condition, which models Alfvén wave propagating along a magnetic field in cold plasmas. We are interested in the double-pole solutions that are a kind of bound-state solitons. By solving a direct scattering problem, the Jost eigenfunctions and scattering matrix are given, their symmetries and asymptotic behaviors are also presented. Then the inverse scattering problems are solved in terms of the matrix Riemann–Hilbert method. In addition, the trace formulae for analytic scattering coefficients and theta conditions are derived. Finally, the explicit formulae of double-pole solutions for the equation are obtained.

Riemann–Hilbert problems and soliton solutions for a generalized coupled Sasa–Satsuma equation

This paper studies the multi-component Sasa-Satsuma integrable hierarchies via an arbitrary-order matrix spectral problem, based on the zero curvature formulation. A generalized coupled Sasa-Satsuma equation is derived from the multi-component Sasa-Satsuma integrable hierarchies with a bi-Hamiltonian structure. The inverse scattering transform of the generalized coupled Sasa-Satsuma equation is presented by the spatial matrix spectral problem and the Riemann–Hilbert method, which enables us to obtain the N-soliton solutions. And then the dynamics of one- and two-soliton solutions are discussed and presented graphically. Asymptotic analyses of the presented two-soliton solution are finally analyzed.

Long-time asymptotics of the good Boussinesq equation with q xx-term and its modified version

Two modified Boussinesq equations along with their Lax pairs are proposed by introducing the Miura transformations. The modified good Boussinesq equation with initial condition is investigated by the Riemann–Hilbert method. Starting with the three-order Lax pair of this equation, the inverse scattering transform is formulated and the Riemann–Hilbert problem is established, and the properties of the reflection coefficients are presented. Then, the formulas of long-time asymptotics to the good Boussinesq equation and its modified version are given based on the Deift–Zhou approach of nonlinear steepest descent analysis. It is demonstrated that the results from the long-time asymptotic analysis are in excellent agreement with the numerical solutions. This is the first result on the long-time asymptotic behaviors of the good Boussinesq equation with qxx-term and its modified version.

Holomorphic solutions and solvability theory for a class of linear complete singular integro-differential equations with convolution by Riemann–Hilbert method

Holomorphic solutions and solvability theory for a class of linear complete singular integro-differential equations with convolution by Riemann–Hilbert method

Application of Riemann–Hilbert method to an extended coupled nonlinear Schrödinger equations

Based on the non-semisimple Lie algebra g˜, we introduce a new integrable coupling system of the focusing nonlinear Schrödinger equation associated with a 4th-order block matrix spectral problem, which is called the extended coupled nonlinear Schrödinger (ECNLS) equations. The ECNLS equations can be reduced to many equations with physical background, including the focusing nonlinear Schrödinger equations (NLS) equation, the new integrable system of coupled NLS equations, the Manakov system, the mixed coupled NLS equations, etc. Based on the 4th-order spectral problem, we analyze the asymptotic behavior, analyticity and symmetry of the eigenfunctions and scattering coefficients. It follows that a formulation of solutions is developed for the Riemann–Hilbert problems associated with the reflectionless transforms. Furthermore, the N-soliton solutions of the ECNLS equations are generated. Additionally, the ECNLS equations are extended to a multi-component nonlinear Schrödinger system by means of a new N×N non-semisimple Lie algebra ĝ, which means that the ECNLS equations are extended to an arbitrary number of components. Actually, the coupled and multi-component equations that we obtained can enrich the existing integrable models and possibly describe new nonlinear phenomena.

Formula omitted]-(soliton, breather) interactions for general multi-component third-fifth-order mKdV equations via Riemann–Hilbert method

In this paper, we mainly focus on the Riemann–Hilbert approach solving the general n-component third-fifth-order mKdV (n-(3,5)-mKdV) equations containing the n-component mKdV equation, fifth-order mKdV equation, and their combination. Starting from the spectral analysis of the (n+1)-order matrix Lax pair, we give the corresponding (n+1)-order matrixed-type Riemann–Hilbert problem. By solving the Riemann–Hilbert problem, N-soliton solutions of the n-(3,5)-mKdV equations can be found. Particularly, for the case of reflectionless and some types of spectral parameters of the Lax pair, we analyze the interactions of some kinds of solutions of the 3-component mKdV equation, fifth-order mKdV equation, and third-fifth-order mKdV equations, including breather solutions, W-shaped solutions, anti-bright and bright solutions. Some elastic collisions between them are also presented. In addition, the multi-pole solutions are obtained for the n-(3,5)-mKdV equations through the L’Hospital’s rule. These results will be useful to further understand the related wave phenomena in the multi-component physical systems.

N-soliton solutions for the coupled extended modified KdV equations via Riemann–Hilbert approach

In this paper, we utilize the Riemann–Hilbert method to construct the N-soliton solutions for the coupled extended modified KdV equations with zero boundary condition. Firstly, we establish the relationship between the solutions of the coupled extended modified KdV equations and a Riemann–Hilbert problem. Secondly, by solving the particular Riemann–Hilbert problem with an identity jump matrix, the general N-soliton solutions of the coupled extended modified KdV equations are derived. In addition, the local structures of the N-soliton solution are graphically presented by choosing parameters suitably.

Formula omitted]-double poles solutions for nonlocal Hirota equation with nonzero boundary conditions using Riemann–Hilbert method and PINN algorithm

In this paper, we systematically investigate the nonlocal Hirota equation with nonzero boundary conditions via Riemann–Hilbert method and multi-layer physics-informed neural networks algorithm. Starting from the Lax pair of nonzero nonlocal Hirota equation, we first give out the Jost function, scattering matrix, their symmetry and asymptotic behavior. Then, the Riemann–Hilbert problem with nonzero boundary conditions are constructed and the precise formulae of N-double poles solutions and N-simple poles solutions are written by determinants. Different from the local Hirota equation, the symmetry of scattering data for nonlocal Hirota equation is completely different, which results in disparate discrete spectral distribution. In particular, it could be more complicated and difficult to obtain the symmetry of scattering data under the circumstance of double poles. Besides, we also analyze the asymptotic state of one-double poles solution as t→∞. Whereafter, the multi-layer physics-informed neural networks algorithm is applied to research the data-driven soliton solutions of the nonzero nonlocal Hirota equation by using the training data obtained from the Riemann–Hilbert method. Most strikingly, the integrable nonlocal equation is firstly solved via multi-layer physics-informed neural networks algorithm. As we all know, the nonlocal equations contain the PT symmetry P:x→−x, or T:t→−t, which are different with local ones. Adding the nonlocal term into the neural network, we can successfully solve the integrable nonlocal Hirota equation by multi-layer physics-informed neural networks algorithm. The numerical results show that the algorithm can recover the data-driven soliton solutions of the integrable nonlocal equation well. Noteworthily, the inverse problems of the integrable nonlocal equation are discussed for the first time through applying the physics-informed neural networks algorithm to discover the parameters of the equation in terms of its soliton solution.

N-soliton solutions of the generalized mixed nonlinear Schrödinger equation through the Riemann–Hilbert method

The generalized mixed nonlinear Schrödinger equation is the core of this thesis study and it plays a pivotal position in many physical applications. In this paper, on the whole, in the framework of the Riemann–Hilbert approach, by analyzing the spectral problem of the Lax pair and applying the transformation of the matrix, the Riemann–Hilbert problem of the generalized mixed nonlinear Schrödinger equation for specific values of the parameters is constructed. After that, in the case of irregularity, the [Formula: see text]-soliton solutions of this equation at specific values of the parameters can be presented by the scattering transformation. In particular, some three-dimensional images of this nonlinear system can be graphically depicted.

Riemann-Hilbert method and multi-soliton solutions of an extended modified Korteweg-de Vries equation with N distinct arbitrary-order poles

The Riemann-Hilbert (RH) method is developed to study the extended modified Korteweg-de Vries (emKdV) equation with zero boundary conditions. The analytical and asymptotic properties of Jost functions are obtained by the direct scattering analysis to establish a suitable RH problem. We consider the singular RH problem of scattering data with N distinct poles. By using the generalized residue condition to solve the RH problem, we construct the exact solution of emKdV equation under the condition of no reflection. In addition, four kinds of special poles and their corresponding soliton solutions are discussed in detail, including one second-order pole, one third-order pole, three first-order poles and two second-order poles.

On smooth perturbations of Chebyshëv polynomials and -Riemann–Hilbert method

Abstract $\bar {\partial } $ -extension of the matrix Riemann–Hilbert method is used to study asymptotics of the polynomials $ P_n(z) $ satisfying orthogonality relations $$ \begin{align*} \int_{-1}^1 x^lP_n(x)\frac{\rho(x)dx}{\sqrt{1-x^2}}=0, \quad l\in\{0,\ldots,n-1\}, \end{align*} $$ where $ \rho (x) $ is a positive $ m $ times continuously differentiable function on $ [-1,1] $ , $ m\geq 3 $ .

Riemann–Hilbert problem and interactions of solitons in the ‐component nonlinear Schrödinger equations

AbstractIn this work, we systematically investigate the general‐component nonlinear Schrödinger equations by using the Riemann–Hilbert method. Starting from the Lax pair associated with anmatrix spectral problem, we construct a Riemann–Hilbert problem and obtain the compact‐soliton solutions formula expressed by determinants. Taking two‐components nonlinear Schrödinger equations as an example, this work detailedly analyzes its multisoliton solutions by means of parameters modulation. Many interesting new phenomena are displayed including elastic collision, soliton reflection, parallel propagation, time‐periodic propagation and (space, time)‐periodic propagation. In addition, the interactions between solitons from weak to strong are presented. Finally, some conjectures about the dynamical behaviors of‐soliton solutions are proposed. We hope that the present analysis would be useful for a better understanding of soliton interactions in related fields such as nonlinear optics, plasma physics, oceanography, and Bose–Einstein condensates.

Analysis for an infinite orthotropic elastic plane with a hole subjected to uniform heat flux

A general solution is derived for the thermal stresses of an infinite orthotropic plane with a hole subjected to uniform heat flux. A dislocation method is applied to solve uniform heat flux problem, in which the orthotropic thermal conductivities and coefficients of thermal expansion are considered. The dislocation functions for stress analysis are also considered. The dislocations of displacements for the x and y directions due to the thermal stress and the elastic stress at infinity must be zero. Thus, the dislocation functions regarding elastic stress are determined. Then, the stress analysis is carried out by two methods, that is, methods due to the Riemann-Hilbert method and Cauchy integral method. These methods are completely different. It is confirmed that the stress functions achieved are identical. The final exact stress functions as a general solution are derived as a closed form. The stress components are represented by one complex variable. Therefore, the calculation of the stress components is simple. Arbitrarily shaped hole problem can be solved by exchanging the mapping function. As an example, an infinite plane with a square hole subjected to uniform heat flux is analyzed. The heat flux and stress distributions are shown.

Riemann–Hilbert problem for the generalized modified Korteweg–de Vries equation with N distinct arbitrary-order poles

In this work, the generalized modified Korteweg–de Vries (gmKdV) equation is constructed by the first time and is solved by the Riemann–Hilbert method with the zero boundary condition. In the direct scattering transform, the analytical and asymptotic properties related to the Jost functions and the scattering matrix are given. On the basis of the above results, the appropriate Riemann–Hilbert problem (RHP) is constructed. By solving the RHP, we obtain the exact solution of the gmKdV equation in the case of no reflection potential when the scattering data [Formula: see text] has simple poles and higher-order poles. Furthermore, the three special solutions under different zero points are given and the phenomenon of their spread is described, respectively.

Stress analysis for an orthotropic elastic half plane with an oblique edge crack and stress intensity factors

The main purposes of the present paper are as follows: (1) to analytically derive the general solution (stress functions) for an orthotropic elastic half plane with a crack or a notch; (2) to derive the Riemann–Hilbert equation as the analytical method; (3) to solve the present problem using two methods, one is a Cauchy integral method and other is a Riemann–Hilbert method; and (4) to derive the general expressions of the stress intensity factor (SIF) for a crack problem. The stress functions obtained by the Riemann–Hilbert equation are compared with those obtained by Cauchy integral method. Then, it is confirmed that the same stress functions can be obtained. The stress functions are expressed by any elastic constants. Therefore, Mode I and II SIFs can be calculated for any elastic constants. Some examples are shown. It is stated from the results of the examples that the negative Mode I SIFs exist for certain oblique edge crack angles and elastic constants. Because the derived stress functions are expressed using a mapping function, other geometrical shapes can be analyzed by changing the mapping function.

Riemann–Hilbert method for multi-soliton solutions of a fifth-order nonlinear Schrödinger equation

A fifth-order nonlinear Schrodinger equation which describes the one-dimensional anisotropic Heisenberg ferromagnetic spin chain is under investigation in this paper. Starting from the spectral analysis, a matrix Riemann–Hilbert problem is established on the real axis. Then, through solving the resulting matrix Riemann–Hilbert problem under the condition of no reflection, we systematically derive multi-soliton solutions to the fifth-order nonlinear Schrodinger equation. In addition, the localized structures of one-soliton solution are shown vividly via a few plots.

Riemann–Hilbert method and multi-soliton solutions of the Kundu-nonlinear Schrödinger equation

In this work, we study the Kundu-nonlinear Schrodinger (Kundu-NLS) equation (so-called the extended NLS equation), which can describe the propagation of the waves in dispersive media. A Lax spectral problem is used to construct the Riemann–Hilbert problem, via a matrix transformation. Based on the inverse scattering transformation, the general solutions of the Kundu-NLS equation are calculated. In the reflection-less case, the special matrix Riemann–Hilbert problem is carefully proposed to derive the multi-soliton solutions. Finally, some novel dynamics behaviors of the nonlinear system are theoretically and graphically discussed.

Entanglement entropy of two disjoint intervals separated by one spin in a chain of free fermion**Gehér was supported by the Leverhulme Trust Early Career Fellowship (ECF-2018-125), and by the Hungarian National Research, Development and Innovation Office (Grant No. K115383). Its was supported by the NSF grant DMS-1700261. Virtanen was supported in part by EPSRC grants EP/M024784/1 and EP/T008636/1. Gehér and

We calculate the entanglement entropy of a non-contiguous subsystem of a chain of free fermions. The starting point is a formula suggested by Jin and Korepin, arXiv:1104.1004, for the reduced density of states of two disjoint intervals with lattice sites P = {1, 2, …, m} ∪ {2m + 1, 2m + 2, …, 3m}, which applies to this model. As a first step in the asymptotic analysis of this system, we consider its simplification to two disjoint intervals separated just by one site, and we rigorously calculate the mutual information between these two blocks and the rest of the chain. In order to compute the entropy we need to study the asymptotic behaviour of an inverse Toeplitz matrix with Fisher–Hartwig symbol using the the Riemann–Hilbert method.

Inverse Scattering Transform and Soliton Classification of Higher-Order Nonlinear Schrödinger-Maxwell-Bloch Equations

We investigate higher-order nonlinear Schrodinger-Maxwell-Bloch equations using the Riemann-Hilbert method. We perform a spectral analysis of the Lax pair and construct a Riemann-Hilbert problem according to the spectral analysis. As a result, we obtain three types of multisoliton solutions. Based on the analytic solution and with a choice of corresponding parameter values, we obtain solutions of the breather type and a bell-shaped solution and find an interesting phenomenon of the collision of two soliton solutions. We hope that these results can be useful in modeling the wave propagation of a nonlinear optical field in an erbium-doped fiber medium.