Wronskian and Grammian Determinant Solutions for a Variable-Coefficient Kadomtsev-Petviashvili Equation

In this paper, we derive the bilinear form for a variable-coefficient Kadomtsev Petviashvili-typed equation. Based on the bilinear form, we obtain the Wronskian determinant solution, which is proved to be indeed an exact solution of this equation through the Wronskian technique. In addition, we testify that this equation can be reduced to a Jacobi identity by considering its solution as a Grammian determinant by means of Pfaffian derivative formulae.     

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.     

The Wronskian technique for nonlinear evolution equations

The investigation of the exact traveling wave solutions to the nonlinear evolution equations plays an important role in the study of nonlinear physical phenomena. To understand the mechanisms of those physical phenomena, it is necessary to explore their solutions and properties. The Wronskian technique is a powerful tool to construct multi-soliton solutions for many nonlinear evolution equations possessing Hirota bilinear forms. In the process of utilizing the Wronskian technique,the main difficulty lies in the construction of a system of linear differential conditions, which is not unique. In this paper,we give a universal method to construct a system of linear differential conditions.     

Generalized Wronskian Solutions to Differential-Difference KP Equation

A new method for constructing the Wronskian entries is proposed and applied to the differential-difference Kadomtsev-Petviashvilli (DΔKP) equation. The generalized Wronskian solutions to it are obtained, including rational solutions and Matveev solutions.     

Quasi-Periodic Solutions and Asymptotic Properties for the Isospectral BKP Equation

In this paper, based on a Riemann theta function and Hirota's bilinear form, a straightforward way is presented to explicitly construct Riemann theta functions periodic waves solutions of the isospectral BKP equation.Once the bilinear form of an equation obtained, its periodic wave solutions can be directly obtained by means of an unified theta function formula and the way of obtaining the bilinear form is given in this paper. Based on this, the Riemann theta function periodic wave solutions and soliton solutions are presented. The relations between the periodic wave solutions and soliton solutions are strictly established and asymptotic behaviors of the Riemann theta function periodic waves are analyzed by a limiting procedure. The N-soliton solutions of isospectral BKP equation are presented with its detailed proof.     

Darboux Transformation and Grammian Solutions for Nonisospectral Modified Kadomtsev-Petviashvili Equation with Symbolic Computation

In the present paper, under investigation is a nonisospectral modified Kadomtsev-Petviashvili equation, which is shown to have two Painlevé branches through the Painlevé analysis. With symbolic computation, two Lax pairs for such an equation are derived by applying the generalized singular manifold method. Furthermore, based on the two obtained Lax pairs, the binary Darboux transformation is constructed and then the N-th-iterated potential transformation formula in the form of Grammian is also presented.     

Quasi-Periodic Solutions and Asymptotic Properties for the Isospectral BKP Equation

In this paper, based on a Riemann theta function and Hirota's bilinear form, a straightforward way is presented to explicitly construct Riemann theta functions periodic waves solutions of the isospectral BKP equation. Once the bilinear form of an equation obtained, its periodic wave solutions can be directly obtained by means of an unified theta function formula and the way of obtaining the bilinear form is given in this paper. Based on this, the Riemann theta function periodic wave solutions and soliton solutions are presented. The relations between the periodic wave solutions and soliton solutions are strictly established and asymptotic behaviors of the Riemann theta function periodic waves are analyzed by a limiting procedure. The N-soliton solutions of isospectral BKP equation are presented with its detailed proof.     

非线性发展方程的Wronskian解及Young图证明

给出一般非线性发展方程构造Wronskian解的间接法. 根据Young图运算的性质给出了文中命题的证明, 并讨论了置换群特征标与Young图表达式系数间的关系. 关键词： 非线性发展方程 Wronskian解 Young图 特征标     

Novel Wronskian Condition and New Exact Solutions to a (3+1)-Dimensional Generalized KP Equation

Utilizing the Wronskian technique, a combined Wronskian condition is established for a (3+1)-dimensional generalized KP equation. The generating functions for matrix entries satisfy a linear system of new partial differential equations. Moreover, as applications, examples of Wronskian determinant solutions, including N-soliton solutions, periodic solutions and rational solutions, are computed.     

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.     

Double Wronskian Solution and Soliton Properties of the Nonisospectral BKP Equation

Based on the Wronskian technique and Lax pair,double Wronskian solution of the nonisospectral BKP equation is presented explicitly.The speed and dynamical influence of the one soliton are discussed.Soliton resonances of two soliton are shown by means of density distributions.Soliton properties are also investigated in the inhomogeneous media.     

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 Form Solutions for a Variable Coefficient Kadomtsev–Petviashvili Equation

Starting from a simple transformation, and with the aid of symbolic computation, we establish the relation- ship between the solution of a generalized variable coefficient Kadomtsev-Petviashvili （vKP） equation and the solution of a system of linear partial differential equations. According to this relation, we obtain Wronskian form solutions of the vKP equation, and further present N-soliton-like solutions for some degenerated forms of the vKP equation. Moreover, we also discuss the influences of arbitrary constants on the soliton and N-soliton solutions of the KPII equation.     

Extended Wronskian Determinant Approach and Iterative Solutions of One-Dimensional Dirac Equation

An approximation method, namely, the Extended Wronskian Determinant Approach, is suggested to study the one-dimensional Dirac equation. An integral equation, which can be solved by iterative procedure to find the wave functions, is established. We employ this approach to study the one-dimensional Dirac equation with one-well potential, and give the energy levels and wave functions up to the first order iterative approximation. For double-well potential, the energy levels up to the first order approximation are given.     

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.     

Generalized Wronskian Solutions to Modified Korteweg-de Vries Equation via Its Backlund Transformation

In the paper we discuss the Wronskian solutions of modified Korteweg-de Vries equation （mKdV） via the Backlund transformation （BT） and a generalized Wronskian condition is given, which allows us to substitute an arbitrary coefficient matrix in the GN （t） for the original diagonal one.     

Linear superposition of Wronskian rational solutions to the KdV equation

A linear superposition is studied for Wronskian rational solutions to the Kd V equation, which include rogue wave solutions. It is proved that it is equivalent to a polynomial identity that an arbitrary linear combination of two Wronskian polynomial solutions with a difference two between the Wronskian orders is again a solution to the bilinear Kd V equation. It is also conjectured that there is no other rational solutions among general linear superpositions of Wronskian rational solutions.     

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.     

Generalized Wronskian Solutions to Modified Korteweg-de Vries Equation via Its Bäcklund Transformation

In the paper we discuss the Wronskian solutions of modified Korteweg-de Vries equation (mKdV) via the Bäcklund transformation (BT) and a generalized Wronskian condition is given, which allows us to substitute an arbitrary coefficient matrix in the G_N(t) for the original diagonal one.     

Grammian Determinant Solution and Pfaffianization for a (3+1)-Dimensional Soliton Equation

Based on the Pfaffian derivative formulae, a Grammian determinant solution for a (3+1)-dimensional soliton equation is obtained. Moreover, the Pfaffianization procedure is applied for the equation to generate a new coupled system. At last, a Gram-type Pfaffian solution to the new coupled system is given.