Generalized double Wronskian solutions of the third-order isospectral AKNS equation

The generalized double Wronskian solutions of the third-order isospectral AKNS equation are obtained. Thus we found rational solutions, Matveev solutions, complexitons and interaction solutions. Moreover, rational solutions of the mKdV equation and KdV equation in double Wronskian form are constructed 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.     

Quasi-periodic solutions of mixed AKNS equations

A hierarchy of new nonlinear evolution equations, which are composed of the positive and negative AKNS flows, is proposed. On the basis of the theory of algebraic curves, the corresponding flows are straightened using the Abel-Jacobi coordinates. The meromorphic function ?, the Baker-Akhiezer vector , and the hyperelliptic curve K_n are introduced and, by using these, quasi-periodic solutions of the first three nonlinear evolution equations in the hierarchy are constructed according to the asymptotic properties and the algebro-geometric characters of ?, and K_n.     

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.     

多分量AKNS方程的新可积分解

从一个任意阶矩阵谱问题出发,多分量AKNS方程的新可积分解被导出.通过利用迹恒等式建立了其双哈密顿结构.同时,证明了空间与时间的约束流在刘维尔意义下是两个完全可积的哈密顿系统.     

Determinant representation of Darboux transformation for the AKNS system

The n-fold Darboux transform (DT) is a 2×2 matrix for the Ablowitz-Kaup-Newell-Segur (AKNS) system. In this paper, each element of this matrix is expressed by 2n 1 ranks' determinants. Using these formulae, the determinant expressions of eigenfunctions generated by the n-fold DT are obtained. Furthermore, we give out the explicit forms of the n-soliton surface of the Nonlinear Schrodinger Equation (NLS) by the determinant of eigenfunctions.     

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.     

两个方程的 Wronskian 解及 Young 图证明

In this paper, we obtain the linear differential conditions of （3 ＋ 1）-dimensional Jimbo-Miwa equation and Boiti-Leon-Manna-Pempinelli equation, which guarantee that the corresponding Wronskian determinant solves the two equations in the Hirota bilinear form. By using the properties of Young diagram, we have proved the results.     

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.     

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).     

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.     

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.

     

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.     

Exact solutions of a negative matrix AKNS system with a Hermitian symmetric space

Based on a 4 × 4 matrix Lax pair, we propose a negative matrix AKNS system with a Hermitian symmetric space. A Darboux transformation is constructed by setting a restrictive condition and using the loop group method. The restrictive condition can guarantee the evolution relations of the potential matrices. Using this Darboux transformation and different seed solutions and free parameters, we obtain different types of spatial–temporal distribution structures for various explicit solutions of the negative matrix AKNS system with a Hermitian symmetric space, including the rogue wave, Ma breather, the interaction of two Ma breathers, and parabolic-type soliton solutions.     

An alternative approach to solve the mixed AKNS equations

The algebraic–geometric solutions of the mixed AKNS equations are investigated through a finite-dimensional Lie–Poisson Hamiltonian system, which is generated by the nonlinearization of the adjoint equation related to the AKNS spectral problem. First, each mixed AKNS equation can be decomposed into two compatible Lie–Poisson Hamiltonian flows. Then the separated variables on the coadjoint orbit are introduced to study these Lie–Poisson Hamiltonian systems. Further, based on the Hamilton–Jacobi theory, the relationship between the action-angle coordinates and the Jacobi-inversion problem is established. In the end, using Riemann–Jacobi inversion, the algebraic–geometric solutions of the first three mixed AKNS equations are obtained.     

A semilinear elliptic equation with double resonance

In this paper, the existence and multiplicity of a class of double resonant semilinear elliptic equations with the Dirichlet boundary value are studied.     

Positons, negatons and complexitons of the mKdV equation with non-uniformity terms

The N-soliton solution of the mKdV equation with non-uniformity terms is obtained through Hirota method and Wronskian technique. We can also derive its positons, negatons and complexitons by a matrix extension of the Wronskian formulation.     

Mechanical theorem proving in the surfaces using the characteristic set method and Wronskian determinant

In this paper, we generalize the method of mechanical theorem proving in curves to prove theorems about surfaces in differential geometry with a mechanical procedure. We improve the classical result on Wronskian determinant, which can be used to decide whether the elements in a partial differential field are linearly dependent over its constant field. Based on Wronskian determinant, we can describe the geometry statements in the surfaces by an algebraic language and then prove them by the characteristic set method.     

Grammian solutions and pfaffianization of a non-isospectral and variable-coefficient Kadomtsev-Petviashvili equation

In this paper, we first present the Grammian determinant solutions to the non-isospectral and variable-coefficient Kadomtsev-Petviashvili (vcKP) equation. Then, by using the pfaffianization procedure of Hirota and Ohta, a new non-isospectral and variable-coefficient integrable coupled system is generated. Moreover, Gramm-type pfaffian solutions of the pfaffianized system are proposed.     

Exact travelling wave solutions for the dissipative (2 + 1)-dimensional AKNS equation

This paper employs the theory of planar dynamical systems and undetermined coefficient method to study travelling wave solutions of the dissipative (2 + 1)-dimensional AKNS equation. By qualitative analysis, global phase portraits of the dynamic system corresponding to the equation are obtained under different parameter conditions. Furthermore, the relations between the properties of travelling wave solutions and the dissipation coefficient r of the equation are investigated. In addition, the possible bell profile solitary wave solution, kink profile solitary wave solutions and approximate damped oscillatory solutions of the equation are obtained by using undetermined coefficient method. Error estimates indicate that the approximate solutions are meaningful. Based on above studies, a main contribution in this paper is to reveal the dissipation effect on travelling wave solutions of the dissipative (2 + 1)-dimensional AKNS equation.