A new method for constructing soliton solutions and periodic solutions of nonlinear evolution equations

With the aid of the symbolic computation, we improve Xie's algorithm [F. Xie, Z.Y. Yan, H. Zhang, Phys. Lett. A 285 (2001) 76], and present a new extended method. Based on the new general ansatz (3), we successfully solve a compound KdV-MKdV equation, and obtain some special solutions which contain soliton solutions, and periodic solutions. The method can also be applied to other nonlinear partial differential equations.     

Double-Alekseev inverse scattering method in the stationary axi-symmetric vacuum gravitation field equations

We present a new improvement to the Alekseev inverse scattering method. This improved inverse scattering method is extended to a double form, followed by the generation of some new solutions of the double-complex Kinnersley equations. As the double-complex function method contains the Kramer-Neugebauer substitution and analytic continuation, a pair of real gravitation soliton solutions of the Einstein’s field equations can be obtained from a double N-soliton solution. In the case of the flat Minkowski space background solution, the general formulas of the new solutions are presented.      

Yang-Mills equations as inverse scattering problem

Non-linear self-duality equations are shown to be conditions of compatibility of two linear equations. All the N-instanton fields are constructed explicitly.     

Exact class of inverse scattering solutions

After presenting a procedure for finding exact solutions to the Gelfand-Levitan equation for inverse scattering, we present explicit solutions for 3- to 6-pole reflection coefficients. The method is applicable to an arbitrarily large number of poles. Here we give the first explicit treatment of 4, 5, and 6 poles.     

Solutions of the anti-self-dualSU(2) Yang-Mills equations gained from the inverse scattering method

By applying the inverse scattering method in connection with a dressing procedure a huge class of solutions of the anti-self-dualSU(2) Yang-Mills equation is obtained. This class contains also the well-known't Hooft instantions.     

Finite-range solutions to the one-dimensional inverse scattering problem

The purpose of this paper is to suggest some ideas concerning the obtention of finite range solutions to the so-called one-dimensional inverse scattering problem. A necessary condition on the ratio of the reflection and transmission coefficients is given for the corresponding potential to be cutoff on both sides.     

Pure soliton solutions of some nonlinear partial differential equations

A general approach is given to obtain the system of ordinary differential equations which determines the pure soliton solutions for the class of generalized Korteweg-de Vries equations (cf. [6]). This approach also leads to a system of ordinary differential equations for the pure soliton solutions of the sine-Gordon equation.     

A Reduced System of Linear Algebraic Equations for Giving Soliton Solutions of the Generalized ZS/AKNS Systems

A simple method for finding soliton solutions of the generaked ZS/AKNS systems whose Lax pairs are matrices with high orders is considered. An explicit expreesion of transformation between the Jost solution relating to the (n-1)-soliton solution and that relating to the n-soliton solution is found. A reduced system of N algebraic equations for giving N soliton solutions is deduced, it has an identical form no matter how high the order of matrices of the Lax pain is.     

On the method of inverse scattering problem and Bäcklund transformations for supersymmetric equations

Supersymmetric Liouville and sine-Gordon equations are studied. We write down for these models the system of linear equations for which the method of inverse scattering should be applicable. Expressions for an infinite set of conserved currents are explicitly given. Supersymmetric Bäcklund transformations and generalized conservation laws are constructed.     

Axially symmetric static solutions of four-dimensional SU(N) σ-models by the inverse scattering method

A systematic framework is derived for constructing a superpotential in static, axially symmetric four-dimensional SU(N) principal σ-models by applying an inverse scattering method.     

Supersymmetric Yang-Mills equations as an inverse scattering problem

The equations of motion (for N=3, 4) and the constraint equations (N=1, 2) for supersymmetric Yang-Mills theories are shown to be the compatibility conditions of some system of linear equations with a parameter.     

On the physical interpretation of some simple soliton solutions of Einstein's equations

The simple soliton solutions of Einstein's equations obtained by Belinski and Zakharov using the inverse scattering method have been interpreted as gravitational (solitary) shock waves, partly on the basis of an analysis of certain (coordinate) singularities apparently inherent to the method. A closer study reveals, however, that such singularities can be removed by an appropriate extension of the solutions, which is given explicitly. A classification of inequivalent flat space-time metrics appropriate for the applications of the method is derived. The problem of matching the Belinski-Zakharov (B-Z) simple solitons to flat space-time is analyzed and found to have more than one solution depending on the type of singularity admitted in the Ricci tensor. This is further illustrated by considering a three-parameter solution, inequivalent to that of Belinski and Zakharov. For negative values of one of these parameters the ranges of the coordinates are limited only by curvature singularities. For positive values of the parameter, coordinate singularities, similar to those in the B-Z solution, are also present. In this case, however, a matching to flat space-time leads to a shock front whose intersection with any spacelike hypersurface is bounded, in contrast with the behavior displayed by the B-Z solutions. The limiting case when the parameter is zero is found to have some special properties. A smooth extension is also shown to exist.This research was supported through a fellowship from the Consejo Nacional de Investigaciones Cientificas y Technicas de la Republica Argentina.     

Singular and non-topological soliton solutions for nonlinear fractional differential equations

In this article, the fractional derivatives are described in the modified Riemann–Liouville sense. We propose a new approach, namely an ansatz method, for solving fractional differential equations(FDEs) based on a fractional complex transform and apply it to solve nonlinear space–time fractional equations. As a result, the non-topological as well as the singular soliton solutions are obtained. This method can be suitable and more powerful for solving other kinds of nonlinear fractional FDEs arising in mathematical physics.     

Quantum inverse scattering method and the Heisenberg ferromagnet

The quantum version of the inverse scattering method is formulated for the Heisenberg ferromagnet. The elements of the transition matrix for the corresponding auxiliary linear problem are the quantum analogue of the action-angle variables. In a continuous limit the model is shown to yield the quantum nonlinear Schrödinger equation.     

Numerical evidence for breakdown of soliton behaviour in solutions of the Maxwell-Bloch equations

Some results of a numerical study of colliding solitary wave solutions of the Maxwell-Bloch equations are reported. At high densities these pulses do not behave as solitons: on collision they undergo deformations and throw off oscillating tails.     

On axially symmetric soliton solutions of the coupled scalar-vector-tensor equations in general relativity

The inverse scattering theory used by Belinsky and Zakharov to obtain soliton solutions of the of the Einstein equations is here applied to the case of a five-dimensional space and interpreted in the framework of the Jordan-Kaluza-Klein theory. For two solitons exact, stationary, axially symmetric and asymptotically flat solutions are obtained.     

Multiple soliton solutions for the integrable couplings of the KdV and the KP equations

In this work, we study the nonlinear integrable couplings of the KdV and the Kadomtsev-Petviashvili (KP) equations. The simplified Hirota’s method will be used for this study. We show that these couplings possess multiple soliton solutions the same as the multiple soliton solutions of the KdV and the KP equations, but differ only in the coefficients of the transformation used. This difference exhibits soliton solutions for some equations and anti-soliton solutions for others.     

On relativistic-invariant formulation of the inverse scattering transform method

A manifestly relativistic-invariant formulation of the method of inverse scattering transform for relativistic-invariant equations is proposed. The sine-Gordon model and the massive Thirring model are considered.     

One and two soliton solutions for seventh-order Caudrey-Dodd-Gibbon and Caudrey-Dodd-Gibbon-KP equations

In this work, we explore more applications of the simplified form of the bilinear method to the seventhorder Caudrey-Dodd-Gibbon (CDG) and the Caudrey-Dodd-Gibbon-KP (CDG-KP) equation. We formally derive one and two soliton solutions for each equation. We also show that the two equations do not show resonance.