Two-component generalized Ragnisco–Tu equation and the Riemann–Hilbert problem

In this paper, we present the inverse scattering transformation (IST) for discrete sine‐Gordon equation via the Riemann–Hilbert approach. In the direct scattering part, we establish analyticity, symmetries, and asymptotic properties of Jost solutions and reflection coefficients. In the inverse part, we solve the discrete sine‐Gordon equation by establishing a Riemann–Hilbert problem with simple and double poles, respectively. N‐soliton solutions are obtained via the reconstruction formula correspondent to the Riemann–Hilbert problem. As examples of N‐solitons, we present some explicit solutions such as breathers, degenerate solitons for discrete sine‐Gordon equation, and the dynamical properties of these solutions are further analyzed.

In this paper, we study the [Formula: see text]-soliton solutions for the Hirota and Maxwell–Bloch equation with physical meaning. From the Lax pair and Volterra integral equations, the Riemann–Hilbert problem of this integrable equation is constructed. By solving the matrix Riemann–Hilbert problem with the condition of no reflecting, the [Formula: see text]-soliton solutions for the Hirota and Maxwell–Bloch equation are obtained explicitly. Finally, we simulate the three-dimensional diagram of [Formula: see text] with 2-soliton solutions and the motion trajectory of [Formula: see text]-axis in the case of different [Formula: see text].

Riemann-Hilbert problems for the Ernst equation and fibre bundles

The main objective of this work is to study the multi-soliton solutions for the higher-order coupled nonlinear Schrodinger system in an optical fiber. Firstly, by using the Riemann–Hilbert (RH) approach as well as analyzing the related spectral problem of the Lax pair, a matrix RH problem for the system is strictly formulated. Secondly, a series of multi-soliton solutions including breathers and interaction solutions can be computed from the RH problem with the reflectionless case. Thirdly, the propagation and collision dynamic behaviors as well as localized wave characteristics of these solutions are presented by selecting appropriate parameters with some graphics. The innovation and highlights of this article are shown through obtained interesting results. The one is that the higher-order linear and nonlinear term $$\varepsilon $$ has important impact on the velocity, phase, period, and wavewidth of wave dynamics. The other is that collisions for the second-order breathers and soliton solutions are elastic interaction which imply they remain bounded all the time. Nevertheless, third-order breathers and soliton solutions are inelastic interaction and the amplitude decreases rapidly with time when collisions occur.

This article’s purpose is to investigate multiple high-order pole solutions for the AB system by the Riemann–Hilbert (RH) approach. We establish the RH problem through using spectral analysis to the Lax pair. Then the RH problem can be resolved and the soliton solution’s formula can be given by using the Laurent expansion method. Finally, we get special soliton solutions, including dark solitons, W-type dark solitons and multiple high-pole solutions. In addition, the W-type dark soliton solutions will occur when the spectral parameters are purely imaginary.

This paper studies the N-soliton solutions of the generalized nonlinear Schrödinger equation. Firstly, the spectral analysis of a Lax pair is carried out. Secondly, the Riemann–Hilbert (RH) problem is established. Finally, the RH problem without reflection is considered and the expression of general N-soliton solutions of the equation is obtained.

We consider matrix orthogonal polynomials related to Bessel type matrices of weights that can be defined in terms of a given matrix Pearson equation. From a Riemann–Hilbert problem we derive first and second order differential relations for the matrix orthogonal polynomials and functions of second kind. It is shown that the corresponding matrix recurrence coefficients satisfy a non-Abelian extension of a family of discrete Painlevé d-PIV equations. We present some nontrivial examples of matrix orthogonal polynomials of Bessel type.

Exact multi-soliton solutions of the KdV equation with a source: Riemann–Hilbert formulation

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.

We investigate the inverse scattering transform for the Schrödinger-type equation under zero boundary conditions with the Riemann–Hilbert (RH) approach. In the direct scattering process, the properties are given, such as Jost solutions, asymptotic behaviors, analyticity, the symmetries of the Jost solutions and the corresponding spectral matrix. In the inverse scattering process, the matrix RH problem is constructed for this integrable equation base on analyzing the spectral problem. Then, the reconstruction formula of potential and trace formula are also derived correspondingly. Thus, N double-pole solutions of the nonlinear Schrödinger-type equation are obtained by solving the RH problems corresponding to the reflectionless cases. Furthermore, we present a single double-pole solution by taking some parameters, and it is analyzed in detail.

The Riemann-Hilbert approach, in conjunction with the canonical Wiener-Hopf factorisation of certain matrix functions called monodromy matrices, enables one to obtain explicit solutions to the non-linear field equations of some gravitational theories. These solutions are encoded in the elements of a matrix M depending on the Weyl coordinates ρ and v, determined by that factorisation. We address here, for the first time, the underlying question of what happens when a canonical Wiener-Hopf factorisation does not exist, using the close connection of Wiener-Hopf factorisation with Toeplitz operators to study this question. For the case of rational monodromy matrices, we prove that the non-existence of a canonical Wiener-Hopf factorisation determines curves in the (ρ, v) plane on which some elements of M(ρ, v) tend to infinity, but where the space-time metric may still be well behaved. In the case of uncharged rotating black holes in four space-time dimensions and, for certain choices of coordinates, in five space-time dimensions, we show that these curves correspond to their ergosurfaces.

In this paper, we study multi-soliton solutions and asymptotic analysis for the coupled variable-coefficient Lakshmanan-Porsezian-Daniel equations, which describe the simultaneous propagation of nonlinear waves in the inhomogeneous optical fibers. We analyze the spectrum of the Lax pair to establish the Riemann-Hilbert problem. Using such Riemann-Hilbert problem, we calculate various multi-soliton solutions without reflection, including breather-like and mixed solitons. We illustrate the propagation and interaction dynamics of the solitons through appropriate parameter selection and asymptotic analysis. We find that the interaction between solitons is elastic, the amplitudes of solitons are only determined by the initial velocity and interaction, and the soliton with lower energy always yields a position shift when elastic interaction occurs. In addition, we observe that the existence time of the wave changes with energy and that multiple elastic interactions between solitons can be obtained when we choose appropriate variable coefficients. Then, we investigate the influences of group velocity dispersions and fourth-order dispersions on the interactions of solitons through parameter modulation mode and asymptotic analysis. Furthermore, we present several new types of nonlinear phenomena graphically, including elastic interactions between parabolic solitons and hump-type solitons, elastic interactions between cubic solitons and hump-type solitons, and periodic-changing propagations.

In this research, the Fokas method is adopted to examine the coupled Gerdjikov-Ivanov equation within the half line interval [Formula: see text]. Meanwhile, the Riemann-Hilbert technique is engaged to work out the potential function associated with the equation. We initially partition the matrix into segments and identify the jump matrix linking each segment based on the positive feature of the segment. The jump matrix comes from the spectral matrix, the latter of which is decided by the initial value and the boundary value. The research indicates that these spectral functions display correlativity instead of being independently separated, and they abide by a global connection while being associated via a compatibility condition. Then, we explore the coupled Gerdjikov-Ivanov equation under the zero boundary condition at infinity. The initial value problem related to the equation is capable of being converted into a Riemann-Hilbert problem on the strength of the analytic and symmetric properties of the eigenfunctions. Ultimately, through the settlement of both the regular and non-regular Riemann-Hilbert problems, a general pattern of N-soliton solutions with respect to the coupled Gerdjikov-Ivanov equation is put forward.

We consider the Riemann-Hilbert factorization approach to the construction of Weyl metrics in four space-time dimensions. We present, for the first time, a rigorous proof of the remarkable fact that the canonical Wiener-Hopf factorization of a matrix obtained from a general (possibly unbounded) monodromy matrix, with respect to an appropriately chosen contour, yields a solution to the non-linear gravitational field equations. This holds regardless of whether the dimensionally reduced metric in two dimensions has Minkowski or Euclidean signature. We show moreover that, by taking advantage of a certain degree of freedom in the choice of the contour, the same monodromy matrix generally yields various distinct solutions to the field equations. Our proof, which fills various gaps in the existing literature, is based on the solution of a second Riemann-Hilbert problem and highlights the deep role of the spectral curve, the normalization condition in the factorization and the choice of the contour. This approach allows us to construct explicit solutions, including new ones, to the non-linear gravitational field equations, using simple complex analytic results.

In this paper, we investigate the Cauchy problem of the Sasa–Satsuma (SS) equation with initial data belonging to the Schwartz space. The SS equation is one of the integrable higher-order extensions of the nonlinear Schrödinger equation and admits a 3 × 3 Lax representation. With the aid of the ∂¯ -nonlinear steepest descent method of the mixed ∂¯ -Riemann–Hilbert problem, we give the soliton resolution and long-time asymptotics for the Cauchy problem of the SS equation with the existence of second-order discrete spectra in the space-time solitonic regions.