广义的带导数非线性薛定谔方程的有理解

广义带导数非线性薛定谔方程是与Kaup-Newell谱问题相联系的一个非线性发展方程,方程可在合适的条件方程下,利用Wronsiki技巧,寻找广义双Wronsikian形式的一般解,进而得到其孤子解和有理解.     

New double Wronskian solutions of the AKNS equation

Soliton solutions, rational solutions, Matveev solutions, complexitons and interaction solutions of the AKNS equation are derived through a matrix method for constructing double Wronskian entries. The latter three solutions are novel. Moreover, rational solutions of the nonlinear Schrodinger equation are obtained by reduction.     

A second Wronskian formulation of the Boussinesq equation

A Wronskian formulation leading to rational solutions is presented for the Boussinesq equation. It involves third-order linear partial differential equations, whose representative systems are systematically solved. The resulting solutions formulas provide a direct but powerful approach for constructing rational solutions, positon solutions and complexiton solutions to the Boussinesq equation. Various examples of exact solutions of those three kinds are computed. The newly presented Wronskian formulation is different from the one previously presented by Li et al., which does not yield rational solutions.     

The generalized Wronskian solutions of a inverse KdV hierarchy

A hierarchy of the inverse KdV equation is discussed. Through the bilinear form of Lax pairs, we prove a generalized Darboux-Crum theorem of the hierarchy. The Bäcklund transformation and the generalized Wronskian solutions are presented. The soliton solutions, explicit rational solutions are obtained then.     

Wronskian determinant solutions of the (3 + 1)-dimensional Jimbo-Miwa equation

A set of sufficient conditions consisting of systems of linear partial differential equations is obtained which guarantees that the Wronskian determinant solves the (3 + 1)-dimensional Jimbo-Miwa equation in the bilinear form. Upon solving the linear conditions, the resulting Wronskian formulations bring solution formulas, which can yield rational solutions, solitons, negatons, positons and interaction solutions.     

2个位势的Ablowitz-Ladik等谱方程的Complexiton解

借助双Casoratian技巧和构造双Wronski行列式元素的矩阵方法,求出2个位势的Ablowitz-Ladik等谱方程的Complexiton解和周期解,并通过将矩阵取成不同的组合类型,进而分别得到该方程具有双Casorati行列式形式的新解,即Complexiton解与类有理解的混合解、Complexiton解与Matveev解的混合解.     

非线性KdV-mKdV方程的新的复合函数解

利用孤立子方程KdV-mKdV的朗斯基解的形式和结构,我们提出了朗斯基形式展开法,运用这一方法获得了KdV-mKdV方程的丰富的新的复合函数解,并且朗斯基行列式中的元素不满足任何线性偏微分方程组.所得到的复合函数解是使用其它的方法得不到的.     

Gauge transformation, elastic and inelastic interactions for the Whitham-Broer-Kaup shallow-water model

Whitham-Broer-Kaup (WBK) model is a model for the dispersive long wave in shallow water. With symbolic computation, gauge transformation between the WBK model and a parameter Ablowitz-Kaup-Newell-Segur (AKNS) system is hereby constructed. By selecting seeds, we derive two sorts of multi-soliton solutions for the WBK model via a N-fold Darboux transformation (DT) of the parameter AKNS system, which are expressed in terms of the Vandermonde-like and double Wronskian determinants, respectively. Different from the bilinear way, the double Wronskian solutions can be obtained via the N-fold DT with a linear algebraic system and matrix differential equation solved. A novel inelastic interaction is graphically discussed, in which the soliton complexes are formed after the collision. Our results could be helpful for interpreting certain shallow-water-wave phenomena.     

Solutions of Nonlocal Equations Reduced from the AKNS Hierarchy

In this paper, nonlocal reductions of the Ablowitz–Kaup–Newell–Suger (AKNS) hierarchy are collected, including the nonlocal nonlinear Schrödinger hierarchy, nonlocal modified Korteweg‐de Vries hierarchy, and nonlocal versions of the sine‐Gordon equation in nonpotential form. A reduction technique for solutions is employed, by which exact solutions in double Wronskian form are obtained for these reduced equations from those double Wronskian solutions of the AKNS hierarchy. As examples of dynamics, we illustrate new interaction of two‐soliton solutions of the reverse‐t nonlinear Schrödinger equation. Although as a single soliton, it is stationary that two solitons travel along completely symmetric trajectories in plane and their amplitudes are affected by phase parameters. Asymptotic analysis is given as demonstration. The approach and relation described in this paper are systematic and general and can be used to other nonlocal equations.     

Novel composite function solutions of the modified KdV equation

A Wronskian form expansion method is proposed to construct novel composite function solutions to the modified Korteweg-de Vries (mKdV) equation. The method takes advantage of the forms and structures of Wronskian solutions to the mKdV equation, and Wronskian entries do not satisfy linear partial differential equations. The method can be automatically carried out in computer algebra (for example, Maple).     

Solving the Korteweg-de Vries equation by its bilinear form: Wronskian solutions

A broad set of sufficient conditions consisting of systems of linear partial differential equations is presented which guarantees that the Wronskian determinant solves the Korteweg-de Vries equation in the bilinear form. A systematical analysis is made for solving the resultant linear systems of second-order and third-order partial differential equations, along with solution formulas for their representative systems. The key technique is to apply variation of parameters in solving the involved non-homogeneous partial differential equations. The obtained solution formulas provide us with a comprehensive approach to construct the existing solutions and many new solutions including rational solutions, solitons, positons, negatons, breathers, complexitons and interaction solutions of the Korteweg-de Vries equation.

     

mKdV-SineGordon方程的多孤子解

该文首先给出了mKdV-SineGordon方程的双线性形式和双线性Backlund变换,然后利用Hirota方法、Backlund变换方法和Wronskian技巧三种不同的方法分别得到mKdV-SineGordon方程的孤子解,最后验证了这三种解的一致性。     

Wronskian and Grammian solutions to a (3 + 1)-dimensional generalized KP equation

Wronskian and Grammian formulations are established for a (3 + 1)-dimensional generalized KP equation, based on the Plücker relation and the Jacobi identity for determinants. Generating functions for matrix entries satisfy a linear system of partial differential equations involving a free parameter. Examples of Wronskian and Grammian solutions are computed and a few particular solutions are plotted.     

The N-soliton solutions of the fifth-order KdV equation under Bargmann constraint

We derive the N-soliton solutions for the fifth-order KdV equation under Bargmann constraint through Hirota method and Wronskian technique, respectively. Some novel determinantal identities and properties are presented to finish the Wronskian verifications. The uniformity of these two kinds of N-soliton solutions is proved.     

Solitons solutions for some nonlinear evolution equations

In this paper, we establish new solitary wave solutions to the modified Kawahara equation by the sine-cosine method. Moreover, the periodic solutions and bell-shaped solitons solutions to the generalized fifth-order KdV equation are obtained. The tanh method is used to handle the double sine-Gordon equation and the double sinh-Gordon equation. Families of exact travelling wave solutions are formally derived. The rational triangle sine-cosine method is introduced and to be constructed complex solutions to the modified Degasperis-Procesi (DP) equation and the modified Camassa-Holm (CH) equation.     

Soliton solutions for a negative order non-isospectral AKNS equation

In this paper, the bilinear form of a negative order non-isospectral AKNS equation is given. The soliton solutions are obtained through Hirota’s direct method and the Wronskian technique, respectively.     

Determinant Solutions Generalized KP Equation to a （3＋1）-Dimensional with Variable Coefficients

A system of linear conditions is presented for Wronskian and Grammian solutions to a (3+1)-dimensional generalized vcKP equation.The formulations of these solutions require a constraint on variable coefficients.     

On intermediate solutions and the Wronskian for half-linear differential equations

The asymptotic behavior of nonoscillatory solutions of the half-linear differential equation is studied. In particular, two Wronskian-type functions, which have some interesting properties, similar to the one of the Wronskian in the linear case, are given. Using these properties and suitable integral inequalities, the existence of the so-called intermediate solutions is examined and an open problem is solved.     

用试探方程法求Jaulent-Miodek方程的新的精确行波解

利用试探方程法将Jaulent-Miodek方程约化为初等积分的形式,进而求出了该方程的精确行波解,其中包括椭圆函数双周期解和有理函数解等新解.     

Integrability and exact solutions of the nonautonomous mixed mKdV–sinh–Gordon equation

In this paper, a nonautonomous mixed mKdV–sinh–Gordon equation with one arbitrary time-dependent variable coefficient is discussed in detail. It is proved that the equation passes the Painlevé test in the case of positive and negative resonances, respectively. Furthermore, a dependent variable transformation is introduced to get its bilinear form. Then, soliton, negaton, positon and interaction solutions are introduced by means of the Wronskian representation. Velocities are found to depend on the time-dependent variable coefficient appearing in the equation and this leads to a wide range of interesting behaviours. The singularities and asymptotic estimate of these solutions are discussed. At last, the superposition formulae for these solutions are also constructed.