Rational and Periodic Solutions for a （2＋1）-Dimensional Breaking Soliton Equation Associated with ZS-AKNS Hierarchy

The double Wronskian solutions whose entries satisfy matrix equation for a （2＋1）-dimensional breaking soliton equation （（2＋ 1）DBSE） associated with the ZS-AKNS hierarchy are derived through the Wronskian technique. Rational and periodic solutions for （2＋1）DBSE are obtained by taking special eases in general double Wronskian solutions.     

N-Soliton Solution in Wronskian Form for a Generalized Variable-Coefficient Korteweg--de Vries Equation

We concentrate on finding exact solutions for a generalized variable-coefficient Korteweg-de Vries equation of physically significance. The analytic N-soliton solution in Wronskian form for such a model is postulated and verified by direct substituting the solution into the bilinear form by virtue of the Wronskian technique. Additionally, the bilinear auto-Backlund transformation between the （ N - 1）- and N-soliton solutions is verified.     

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-Backlund transformation for a variable-coemcient Korteweg-de Vries equation with nonuniformities are given. Then, the N-solitonic solution in terms of Wronskian form is obtained and verified. In addition, it is shown that the （N - 1）- and N-solitonic solutions satisfy the auto-Backlund transformation through the Wronskian technique.     

Solutions of the nonlocal(2+1)-D breaking solitons hierarchy and the negative order AKNS hierarchy

The(2+1)-dimensional nonlocal breaking solitons AKNS hierarchy and the nonlocal negative order AKNS hierarchy are presented.Solutions in double Wronskian form of these two hierarchies are derived by means of a reduction technique from those of the unreduced hierarchies.The advantage of our method is that we start from the known solutions of the unreduced bilinear equations,and obtain solitons and multiple-pole solutions for the variety of classical and nonlocal reductions.Dynamical behaviors of some obtained solutions are illustrated.It is remarkable that for some real nonlocal equations,amplitudes of solutions are related to the independent variables that are reversed in the real nonlocal reductions.     

Novel Interaction Solutions to Kadomtsev-Petviashvili Equation

In this paper, based on the forms and structures of Wronskian solutions to soliton equations, a Wronskian form expansion method is presented to find a new class of interaction solutions to the Kadomtsev-Petviashvili equation. One characteristic of the method is that Wronskian entries do not satisfy linear partial differential equation.     

Wronskian and Grammian solutions for the （3＋ 1）-dimensional Jimbo-Miwa equation

<正>In this paper,based on Hirota’s bilinear method,the Wronskian and Grammian techniques,as well as several properties of the determinant,a broad set of sufficient conditions consisting of systems of linear partial differential equations are presented.They guarantee that the Wronskian determinant and the Grammian determinant solve the (3 + 1)-dimensional Jimbo-Miwa equation in the bilinear form.Then some special exact Wronskian and Grammian solutions are obtained by solving the differential conditions.At last,with the aid of Maple,some of these special exact solutions are shown graphically.     

Exact Solutions to a （3＋l）-Dimensional Variable-Coefficient Kadomtsev-Petviashvilli Equation via the Bilinear Method and Wronskian Technique

By truncating the Painleve expansion at the constant level term, the Hirota bilinear form is obtained for a （3＋1）-dimensional variable-coefficient Kadomtsev Petviashvili equation. Based on its bilinear form, solitary-wave solutions are constructed via the ε-expansion method and the corresponding graphical analysis is given. Furthermore, the exact solution in the Wronskian form is presented and proved by direct substitution into the bilinear equation.     

Exact Analytic N-Soliton-Like Solution in Wronskian Form for a Generalized Variable-Coefficient Korteweg-de Vries Model from Plasmas and Fluid Dynamics

Applicable in fluid dynamics and plasmas, a generalized variable-coefficient Korteweg-de Vries （vcKdV） model is investigated. The bilinear form and analytic N-soliton-like solution for such a model are derived by the Hirota method and Wronskian technique. Additionally, the bilinear auto-Bǎcklund transformation between （N-1）- soliton-like and N-soliton-like solutions is verified.     

Double Wronskian Soliton Solution for Mixed AKNS System

The Lax pair of the mixed Ablowitz-Kap-Newell-Segur （AKNS） system is obtained from compatibility condition. Hirota＇s bilinear form is derived by some dependent variable transformation. Moreover, by means of the Wronskian technique, the double Wronskian form of soliton solutions are found. Specially, the two-soliton solution is presented.     

Grammian Solutions to a Variable-Coefficient KP Equation

Solutions in the Grammian form for a variable-coefficient Kadomtsev-Petviashvili （KP） equation which has the Wronskian solutions are derived by means of Pfaffian derivative formulae.     

Double Wronskian Solutions of Non-Isospectral Levi Equations

The double Wronskian solutions of the non-isospectral Levi equationsare derived through Wronskian technique.     

Rational solutions of the general nonlinear Schrödinger equation with derivative

The double Wronskian solutions whose entries satisfy matrix equation of the general nonlinear Schrödinger equation with derivative (GDNLSE) are derived through the Wronskian technique. Soliton solutions and rational solutions of GDNLSE are obtained by taking special cases in general solutions.     

New Double Wronskian Solutions of the Whitham-Broer-Kaup System: Asymptotic Analysis and Resonant Soliton Interactions

In this paper, by the Darboux transformation together with the Wronskian technique, we construct new double Wronskian solutions for the Whitham-Broer-Kaup (WBK) system. Some new determinant identities are developed in the verification of the solutions. Based on analyzing the asymptotic behavior of new double Wronskian functions as t → ±∞, we make a complete characterization of asymptotic solitons for the non-singular, non-trivial and irreducible soliton solutions. It turns out that the solutions are the linear superposition of two fully-resonant multi-soliton configurations, in each of which the amplitudes, velocities and numbers of asymptotic solitons are in general not equal as t → ±∞. To illustrate, we present the figures for several examples of soliton interactions occurring in the WBK system.     

Novel Interaction Solutions to Kadomtsev-Petviashvili Equation

In this paper, based on the forms and structures of Wronskian solutions to soliton equations, a Wronskian form expansion method is presented to find a new class of interaction solutions to the Kadomtsev-Petviashvili equation. One characteristic of the method is that Wronskian entries do not satisfy linear partial differential equation.     

mKdV-sine-Gordon方程丰富的相互作用解

基于mKdV-sine-Gordon方程的Wronsk解的形式和结构,提出了Wronsk形式展开法,通过这一方法求得了该方程的丰富的相互作用解,该方法的主要特征是不要求Wronsk行列式元素满足线性偏微分方程组。     

New interaction solutions to the KdV equation

In this Letter, a few new types of interaction solutions to the KdV equation are obtained through a constructed Wronskian form expansion method. The method takes advantage of the forms and structures of Wronskian solutions to the KdV equation, and the functions used in the Wronskian determinants don't satisfy the systems of linear partial differential equations.     

Generalized Wronskian and Grammian Solutions to a Isospectral B-type Kadomtsev-Petviashvili equation

Generally speaking, the BKP hierarchy which only has Pfaffian solutions. In this paper, based on the Grammian and Wronskian derivative formulae, generalized Wronskian and Grammian determinant solutions are obtained for the isospectral BKP equation (the second member on the BKP hierarchy) in the Hirota bilinear form. Especially, with the help of the properties of the computing of Young diagram, we have first applied Young diagram proved the proposition of this paper. Moreover, by considering the different combinations of the entries in Wronskian, we obtain various types of Wronskian solutions.     

Combined Wronskian solutions to the 2D Toda molecule equation

By combining two pieces of bi-directional Wronskian solutions, molecule solutions in Wronskian form are presented for the finite, semi-infinite and infinite bilinear 2D Toda molecule equations. In the cases of finite and semi-infinite lattices, separated-variable boundary conditions are imposed. The Jacobi identities for determinants are the key tool employed in the solution formulations.     

Wronskian and Grammian solutions for (3+1)-dimensional Jimbo–Miwa equation

In this paper, based on the Hirota's bilinear method, the Wronskian and Grammian technique, as well as several properties of determinant, a broad set of sufficient conditions consisting of systems of linear partial differential equations are presented, which guarantees that the Wronskian determinant and the Grammian determinant solve the (3+1)-dimensional Jimbo-Miwa equation in the bilinear form, and then some special exact Wronskian and Grammian solutions are obtained by solving the differential conditions. At last, with the aid of Maple, some of these special exact solutions are shown graphically.