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For vector nonlocal reverse-space nonlinear Schrödinger equations, a binary Darboux transformation is formulated by using two sets of eigenfunctions and adjoint eigenfunctions. The resulting binary Darboux transformation has been decomposed into an N -fold product of single binary Darboux transformations. An application starting from zero seed potentials generates a class of soliton solutions. [ABSTRACT FROM AUTHOR]

We construct a class of novel reduced nonlocal reverse-spacetime integrable AKNS equations by taking two group reductions of the AKNS matrix spectral problems of arbitrary order. One reduction is local, replacing the spectral parameter with its complex conjugate while the other is nonlocal, replacing the spectral parameter with itself. Then based on the specific distribution of eigenvalues, we compute soliton solutions by using the Riemann-Hilbert problems with the identity jump matrix, where eigenvalues could equal adjoint eigenvalues. [ABSTRACT FROM AUTHOR]

Two simultaneous nonlocal group constraints of the Ablowitz-Kaup-Newell-Segur matrix eigenvalue problems are discussed, of which one constraint changes the eigenvalue parameter into its negative of the complex conjugate and the other constraint does not change the eigenvalue parameter. Under those two constraints, mixed-type nonlocal integrable nonlinear Schrödinger hierarchies are generated. Further, based on specific distributions of eigenvalues and adjoint eigenvalues, a formulation of soliton solutions is established via the corresponding reflection-less generalized Riemann-Hilbert problems, where eigenvalues and adjoint eigenvalues could be equal. [ABSTRACT FROM AUTHOR]

The paper aims to generate nonlocal integrable nonlinear Schrödinger hierarchies of type (− λ , λ) by imposing two nonlocal matrix restrictions of the AKNS matrix characteristic-value problems of arbitrary order. Based on the explored outspreading of characteristic-values and adjoint characteristic-values, exact soliton solutions are formulated by applying the associated reflectionless generalized Riemann–Hilbert problems, in which characteristic-values and adjoint characteristic-values could have a nonempty intersection. Illustrative models of the resultant mixed-type nonlocal integrable nonlinear Schrödinger equations are presented. [ABSTRACT FROM AUTHOR]

The aim of the paper is to explore non-local reverse-space matrix non-linear Schrödinger equations and their inverse scattering transforms. Riemann–Hilbert problems are formulated to analyse the inverse scattering problems, and the Sokhotski–Plemelj formula is used to determine Gelfand–Levitan–Marchenko-type integral equations for generalised matrix Jost solutions. Soliton solutions are constructed through the reflectionless transforms associated with poles of the Riemann–Hilbert problems. [ABSTRACT FROM AUTHOR]

Reduced nonlocal matrix integrable modified Korteweg–de Vries (mKdV) hierarchies are presented via taking two transpose-type group reductions in the matrix Ablowitz–Kaup–Newell–Segur (AKNS) spectral problems. One reduction is local, which replaces the spectral parameter λ with its complex conjugate λ ∗ , and the other one is nonlocal, which replaces the spectral parameter λ with its negative complex conjugate − λ ∗ . Riemann–Hilbert problems and thus inverse scattering transforms are formulated from the reduced matrix spectral problems. In view of the specific distribution of eigenvalues and adjoint eigenvalues, soliton solutions are constructed from the reflectionless Riemann–Hilbert problems. [ABSTRACT FROM AUTHOR]

The paper aims to discuss nonlocal reverse-space multicomponent nonlinear Schrödinger equations and their inverse scattering transforms. The inverse scattering problems are analyzed by means of Riemann–Hilbert problems, and Gelfand–Levitan–Marchenko-type integral equations for generalized matrix Jost solutions are determined by the Sokhotski–Plemelj formula. Soliton solutions are generated from the reflectionless transforms associated with zeros of the Riemann–Hilbert problems. [ABSTRACT FROM AUTHOR]

We investigate the solvability of an integrable nonlinear nonlocal reverse-time six-component fourth-order AKNS system generated from a reduced coupled AKNS hierarchy under a reverse-time reduction. Riemann–Hilbert problems will be formulated by using the associated matrix spectral problems, and exact soliton solutions will be derived from the reflectionless case corresponding to an identity jump matrix. [ABSTRACT FROM AUTHOR]

The aim of the paper is to construct nonlocal reverse‐space nonlinear Schrödinger (NLS) hierarchies through nonlocal group reductions of eigenvalue problems and generate their inverse scattering transforms and soliton solutions. The inverse scattering problems are formulated by Riemann‐Hilbert problems which determine generalized matrix Jost eigenfunctions. The Sokhotski‐Plemelj formula is used to transform the Riemann‐Hilbert problems into Gelfand‐Levitan‐Marchenko type integral equations. A solution formulation to special Riemann‐Hilbert problems with the identity jump matrix, corresponding to the reflectionless transforms, is presented and applied to N‐soliton solutions of the nonlocal NLS hierarchies. [ABSTRACT FROM AUTHOR]

Abstract A class of Riemann–Hilbert problems on the real axis is formulated for solving the multicomponent AKNS integrable hierarchies associated with a kind of bock matrix spectral problems. Through special Riemann–Hilbert problems where a jump matrix is taken to be the identity matrix, soliton solutions to all integrable equations in each hierarchy are explicitly computed. A class of specific reductions of the presented integrable hierarchies is also made, together with its reduced Lax pairs and soliton solutions. [ABSTRACT FROM AUTHOR]

An arbitrary order matrix spectral problem is introduced and its associated multicomponent AKNS integrable hierarchy is constructed. Based on this matrix spectral problem, a kind of Riemann‐Hilbert problems is formulated for a multicomponent mKdV system in the resulting AKNS integrable hierarchy. Through special corresponding Riemann‐Hilbert problems with an identity jump matrix, soliton solutions to the presented multicomponent mKdV system are explicitly worked out. A specific reduction of the multicomponent mKdV system is made, together with its reduced Lax pair and soliton solutions. [ABSTRACT FROM AUTHOR]

Abstract The inverse scattering transform is developed for a combined modified Korteweg–de Vrie equation through the technique of Riemann–Hilbert problems. From special Riemann–Hilbert problems with an identity jump matrix, soliton solutions are generated, which corresponds to the inverse scattering problems with reflectionless coefficients. A specific example of two-soliton solutions is explicitly presented, together with its 3d plots, contour plots and x -curve plots. [ABSTRACT FROM AUTHOR]

Abstract Using generalized bilinear equations, we construct a (3+1)-dimensional Jimbo–Miwa-like equation which possesses the same bilinear type as the standard (3+1)-dimensional Jimbo–Miwa equation. Classes of lump-type solutions and interaction solutions between lump-type and kink solutions to the resulting Jimbo–Miwa-like equation are generated through Maple symbolic computation. We discuss the conditions guaranteeing analyticity and positiveness of the solutions. By taking special choices of the involved parameters, 3D plots are presented to illustrate the dynamical features of the solutions. [ABSTRACT FROM AUTHOR]

By using the Hirota bilinear form of the KP equation, twelve classes of lump–kink solutions are presented under the help of symbolic computations with Maple. Analyticity is naturally achieved by taking special choices of the involved parameters to guarantee a positive constant term. A key step in generating lump–kink solutions is to combine quadratic functions and the exponential function in the second-order logarithmic derivative transformation. [ABSTRACT FROM AUTHOR]

Sasa–Satsuma type matrix integrable hierarchies are generated from taking two group reductions of replacing the spectral parameter with its complex conjugate and its negative in the matrix AKNS spectral problems. Based on the Lax pairs and the adjoint lax pairs, Riemann–Hilbert problems and thus inverse scattering transforms are formulated for the resulting Sasa–Satsuma type matrix integrable hierarchies, and their soliton solutions are generated from the associated reflectionless Riemann–Hilbert problems. [ABSTRACT FROM AUTHOR]

A kind of novel reduced nonlocal integrable mKdV equations of odd order is presented by taking two group reductions of the AKNS matrix spectral problems. One reduction is local, replacing the spectral parameter with its negative and the other is nonlocal, replacing the spectral parameter with its negative complex conjugate. Based on distribution of eigenvalues, soliton solutions are generated from the corresponding reflectionless Riemann–Hilbert problems. [ABSTRACT FROM AUTHOR]

We conduct two nonlocal group reductions of the AKNS matrix spectral problems to generate a class of nonlocal reverse-spacetime integrable mKdV equations. One reduction replaces the spectral parameter with its negative complex conjugate while the other does not change the spectral parameter. Beginning with the specific distribution of eigenvalues, we construct soliton solutions by solving the corresponding generalized Riemann-Hilbert problems with the identity jump matrix, where eigenvalues could equal adjoint eigenvalues. [ABSTRACT FROM AUTHOR]

Abstract: A five-dimensional symmetry algebra consisting of Lie point symmetries is firstly computed for the nonlinear Schrödinger equation, which, together with a reflection invariance, generates two five-parameter solution groups. Three ansätze of transformations are secondly analyzed and used to construct exact solutions to the nonlinear Schrödinger equation. Various examples of exact solutions with constant, trigonometric function type, exponential function type and rational function amplitude are given upon careful analysis. A bifurcation phenomenon in the nonlinear Schrödinger equation is clearly exhibited during the solution process. [Copyright &y& Elsevier]

We would like to propose a kind of nonlocal real reverse-spacetime integrable hierarchies of PT-symmetric matrix AKNS equations through nonlocal symmetry reductions on the potential matrix, and formulate their associated Riemann–Hilbert problems to determine generalized Jost solutions of arbitrary-order matrix spectral problems. The Sokhotski–Plemelj formula is applied in transforming the associated Riemann–Hilbert problems into Gelfand–Levitan–Marchenko type integral equations. The Riemann–Hilbert problems corresponding to the reflectionless case are solved explicitly, where eigenvalues could equal adjoint eigenvalues, and thus, soliton solutions are presented for the resulting nonlocal real reverse-spacetime integrable PT-symmetric matrix AKNS equations. • Matrix integrable hierarchies. • Matrix nonlocal integrable equations. • Innovative group reductions, which keep the existence of infinitely many symmetries and conservation laws. [ABSTRACT FROM AUTHOR]

We would like to analyze a kind of nonlocal reverse-spacetime integrable PT-symmetric multicomponent modified Korteweg-de Vires (mKdV) equations by making a group of nonlocal reductions, and establish their associated Riemann-Hilbert problems which determine generalized Jost solutions of higher-order matrix spectral problems. The Sokhotski-Plemelj formula is used to transform the associated Riemann-Hilbert problems into Gelfand-Levitan-Marchenko type integral equations. The Riemann-Hilbert problems in the reflectionless case are solved explicitly, and the resulting formulation of solutions enables us to present solitons to the nonlocal reverse-spacetime integrable PT-symmetric mKdV equations. [ABSTRACT FROM AUTHOR]

This paper aims to explore nonlocal complex reverse-spacetime modified Korteweg-de Vries (mKdV) hierarchies via nonlocal symmetry reductions of matrix spectral problems and to construct their soliton solutions by the inverse scattering transforms. The corresponding inverse scattering problems are formulated by building the associated Riemann-Hilbert problems. A formulation of solutions to specific Riemann-Hilbert problems, with the jump matrix being the identity matrix, is established, where eigenvalues could equal adjoint eigenvalues, and thus N-soliton solutions to the nonlocal complex reverse-spacetime mKdV hierarchies are obtained from the reflectionless transforms. [ABSTRACT FROM AUTHOR]

The paper deals with the inverse scattering transforms for nonlocal complex reverse-spacetime multicomponent integrable modified Korteweg–de Vries (mKdV) equations. We establish associated Riemann–Hilbert problems and determine their solutions by the Sokhotski–Plemelj formula. The inverse scattering problems consist of Gelfand–Levitan–Marchenko type equations for the generalized matrix Jost solutions and the recovery formula for the potential. When reflection coefficients are zero, the corresponding Riemann–Hilbert problems yield soliton solutions to the nonlocal complex reverse-spacetime mKdV equations. [ABSTRACT FROM AUTHOR]

The paper aims to present the inverse scattering transforms and soliton solutions for nonlocal reverse-time nonlinear Schrödinger equations. The inverse scattering problems are formulated via Riemann–Hilbert problems, and their solutions are determined by the Sokhotski–Plemelj formula, which close the systems for the Jost solutions. Soliton solutions, corresponding to the reflectionless transforms, are generated from zeros and kernel vectors of the Riemann–Hilbert problems with the identity jump matrix. [ABSTRACT FROM AUTHOR]

Abstract: A double sub-equation method is presented for constructing complexiton solutions of nonlinear partial differential equations (PDEs). The main idea of the method is to take full advantage of two solvable ordinary differential equations with different independent variables. With the aid of Maple, one can obtain both complexiton solutions, combining elementary functions and the Jacobi elliptic function solutions, to nonlinear PDEs. Some illustrative equations are investigated by this means and the corresponding complexiton solutions are computed. [Copyright &y& Elsevier]