A Unified and Explicit Construction of N-Soliton Solutions for the Nonlinear Schrfdinger Equation

An explicit N-fold Darboux transformation with multiparameters for nonlinear Schrodinger equation is constructed with the help of its Lax pairs and a reduction technique. According to this Darboux transformation, the solutions of the nonlinear Schrfdinger equation are reduced to solving a linear algebraic system, from which a unified and explicit formulation of N-soliton solutions with multiparameters for the nonlinear Schrfdinger equation is given.

N-soliton Solution for Hirota-Satsuma Equation

In this work,using the Hirota bilinear method,N-soliton solution is obtained for Hirota-Satsuma nonlinear evolution equation:u_t - u_(xxt) - 3u_xu_t + u_x = 0.

N-soliton solutions for the nonlocal two-wave interaction system via the Riemann–Hilbert method

In this paper, a nonlocal two-wave interaction system from the Manakov hierarchy is investigated via the Riemann–Hilbert approach. Based on the spectral analysis of the Lax pair, a Riemann–Hilbert problem for the nonlocal two-wave interaction system is constructed. By discussing the solutions of this Riemann–Hilbert problem in both the regular and nonregular cases, we explicitly present the N-soliton solution formula of the nonlocal two-wave interaction system. Moreover,the dynamical behaviour of the single-soliton solution is shown graphically.

N-soliton Solution for Two Multidimensional Analogues of the m-KdV Equation

Using the Hirota's bilinear method,some new N-soliton solution are presented for two multidimensional analogues of the m-KdV equation wt+wxxx-6w 2 wx+3 2( w x -1 wy+w-x -1 wz)x=0 and wt+wxxx?6w 2 wx+3 2( wwy+wx-x-1 wy)=0 in view of a different treatment.

The Rational Solutions and the Interactions of the N-Soliton Solutions for Boiti-Leon-Manna-Pempinelli-Like Equation

These rational solutions which can be described a kind of algebraic solitary waves which have great potential in applied value in atmosphere and ocean. It has attracted more and more attention recently. In this paper, the generalized bilinear method instead of the Hirota bilinear method is used to obtain the rational solutions to the (2 + 1)-dimensional Boiti-Leon-Manna-Pempinelli-like equation (hereinafter referred to as BLMP equation). Meanwhile, the (2 + 1)-dimensional BLMP-like equation is derived on the basis of the generalized bilinear operators D3,x D3,y and D3,t. And the rational solutions to the (2 + 1)-dimensional BLMP-like equation are obtained successively. Finally, with the help of the N-soliton solutions of the (2 + 1)-dimensional BLMP equation, the interactions of the N-soliton solutions can be derived. The results show that the two soliton still maintained the original waveform after happened collision.

Dynamical interactions between higher-order rogue waves and various forms ofn-soliton solutions(n→∞)of the(2+1)-dimensional ANNV equation

We present new lemmas,theorem and corollaries to construct interactions among higher-order rogue waves,n-periodic waves and n-solitons solutions(n→∞)to the(2+1)-dimensional asymmetric Nizhnik-Novikov-Veselov(ANNV)equation.Several examples for theories are given by choosing definite interactions of the wave solutions for the model.In particular,we exhibit dynamical interactions between a rogue and a cross bright-dark bell wave,a rogue and a cross-bright bell wave,a rogue and a one-,two-,three-,four-periodic wave.In addition,we also present multi-types interactions between a rogue and a periodic cross-bright bell wave,a rogue and a periodic cross-bright-bark bell wave.Finally,we physically explain such interaction solutions of the model in the 3D and density plots.

Bright-Dark Mixed N-Soliton Solution of the Two-Dimensional Maccari System

The general bright-dark mixed N-soliton solution of the two-dimensional Maccari system is obtained with the KP hierarchy reduction method. The dynamics of single and two solitons are discussed in detail. Asymptotic analysis shows that two solitons undergo elastic collision accompanied by a position shift. Furthermore, our analysis on mixed soliton bound states shows that arbitrary higher-order soliton bound states can take place.

N-Soliton Solutions of Modified KdV Equation under Bargmann Constraint

By means of Hirota method,N-soliton solutions of the modified KdV equation under the Bargmannconstraint are obtained through solving the Bargmann constraint and the related Lax pair and conjugate Lax pair ofthe modified KdV equation.

A Simple Approach to Derive a Novel N-Soliton Solution for a (3+1)-Dimensional Nonlinear Evolution Equation

基于 Hirota 双线性的形式，没有采用标准不安技术的一条简单途径，为构造一个新奇 N-soliton 答案为被介绍一(3+1 ) 维的非线性的进化方程。而且，新奇 N-soliton 答案被显示在 Mathematica 的帮助下有反响的行为。

N-Soliton Solutions of(2+1)-Dimensional Non-isospectral AKNS System

The bilinear form of the(2+1)-dimensional non-isospectral AKNS system is derived.Its N-soliton solutionsare obtained by using the Hirota method.As a reduction,a(2+1)-dimensional non-isospectral Schrdinger equation andits N-soliton solutions are constructed.

Wronskian Form of N-Solitonic Solution for a Variable-Coefficient Korteweg-de Vries Equation with Nonuniformities

By the symbolic computation and Hirota method,the bilinear form and an auto-Bcklund transformationfor a variable-coefficient Korteweg-de Vries equation with nonuniformities are given.Then,the N-solitonic solution interms of Wronskian form is obtained and verified.In addition,it is shown that the(N-1)-and N-solitonic solutionssatisfy the auto-Bcklund transformation through the Wronskian technique.

Riemann-Hilbert problems and N-soliton solutions of the nonlocal reverse space-time Chen-Lee-Liu equation

In this paper,the N-soliton solutions to the nonlocal reverse space-time Chen-Lee-Liu equation have been derived.Under the nonlocal symmetry reduction to the matrix spectral problem,the nonlocal reverse space-time Chen-Lee-Liu equation can be obtained.Based on the spectral problem,the specific matrix Riemann-Hilbert problem is constructed for this nonlocal equation.Through solving this associated Riemann-Hilbert problem,the N-soliton solutions to this nonlocal equation can be obtained in the case of the jump matrix as an identity matrix.

Darboux Transformations and N-soliton Solutions of Two (2+1)-Dimensional Nonlinear Equations

Two Darboux transformations of the(2+1)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawaka(CDGKS)equation and(2+1)-dimensional modified Korteweg-de Vries(mKdV) equation are constructed through the Darboux matrix method, respectively. N-soliton solutions of these two equations are presented by applying the Darboux transformations N times. The right-going bright single-soliton solution and interactions of two and three-soliton overtaking collisions of the(2+1)-dimensional CDGKS equation are studied. By choosing different seed solutions, the right-going bright and left-going dark single-soliton solutions, the interactions of two and three-soliton overtaking collisions, and kink soliton solutions of the(2+1)-dimensional mKdV equation are investigated. The results can be used to illustrate the interactions of water waves in shallow water.

Bell Polynomial Approach and N-Soliton Solutions for a Coupled KdV-mKdV System

In fluid dynamics, plasma physics and nonlinear optics, Korteweg-de Vries (KdV)-type equations are used to describe certain phenomena. In this paper, a coupled KdV-modified KdV system is investigated. Based on the Bell polynomials and symbolic computation, the bilinear form of such system is derived, and its analytic N-soliton solutions are constructed through the Hirota method. Two types of multi-soliton interactions are found, one with the reverse of solitonic shapes, and the other, without. Both the two types can be considered elastic. For a pair of solutions to such system, u and v, with the number of solitons N even, the soliton shapes of u stay unvaried while those of v reverse after the interaction; with N odd, the soliton shapes of both u and v keep unchanged after the interaction.

Interaction solutions and localized waves to the(2+1)-dimensional Hirota-Satsuma-Ito equation with variable coefficient

This article investigates the Hirota-Satsuma-Ito equation with variable coefficient using the Hirota bilinear method and the long wave limit method.The equation is proved to be Painlevé integrable by Painlevé analysis.On the basis of the bilinear form,the forms of two-soliton solutions,three-soliton solutions,and four-soliton solutions are studied specifically.The appropriate parameter values are chosen and the corresponding figures are presented.The breather waves solutions,lump solutions,periodic solutions and the interaction of breather waves solutions and soliton solutions,etc.are given.In addition,we also analyze the different effects of the parameters on the figures.The figures of the same set of parameters in different planes are presented to describe the dynamical behavior of solutions.These are important for describing water waves in nature.

Hirota方法求解Boussinesq方程

利用Hirota方法求解(1+1)维和(2+1)维Boussinesq方程,得到其单孤立子解、双孤立子解以及N孤立子解的解析表达式,并通过数值模拟的方法展示了Boussinesq方程双孤子解的相互作用过程.

(3+1)维ZK方程的N孤子解

运用Hirota方法解析求解Zakharov-Kuznetsov(ZK)方程,得到了ZK方程单孤立子解、双孤立子解以及多孤立子解的具体表达式.用三维图形展示了ZK方程双孤子的特殊相互作用过程.

双线性导数方法求解KdV-mKdV混合方程

利用双线性导数方法求解KdV-mKdV混合方程,得到其单孤立子解、双孤立子解以及n孤立子解的解析表达式.运用数学软件对KdV-mKdV混合方程的非线性色散关系进行了分析,并通过波形图像展示了其单孤立子解、双孤立子解的相互作用过程.

一个2+1维变形Boussinesq方程的N孤子解(英文)

研究了一个2 +1维变形Boussinesq非线性发展方程:utt-uxx-uyy-3(u2)xx-uxxxx=0,运用Hirota双线性方法得到它的N孤子解.

负向等谱4位势Ablowitz-Ladik方程的N孤子解(英文)

通过独立变量变换,给出了负向的等谱4位势Ablowitz-Ladik方程的双线性形式,借助Hirota直接方法得到该方程的N孤子解.