Wen-Xiu Ma

A multiple exp-function method for nonlinear differential equations and its application

A multiple exp-function method for exact multiple wave solutions of nonlinear partial differential equations is proposed. The method is oriented towards the ease of use and capability of computer algebra systems and provides a direct and systematic solution procedure that generalizes Hirotas perturbation scheme. With the help of Maple, applying the approach to the (3+1)-dimensional potential-Yu–Toda–Sasa–Fukuyama equation yields exact explicit one-wave, two-wave and three-wave solutions, which include one-soliton, two-soliton and three-soliton type solutions. Two cases with specific values of the involved parameters are plotted for each of the two-wave and three-wave solutions.

Explicit and exact solutions to a Kolmogorov-Petrovskii-Piskunov equation

Abstract Some explicit traveling wave solutions to a Kolmogorov-Petrovskii-Piskunov equation are presented through two ansatze. By a Cole-Hopf transformation, this Kolmogorov-Petrov-skii-Piskunov equation is also written as a bilinear equation and two solutions to describe nonlinear interaction of traveling waves are further generated. Backlund transformations of the linear form and some special cases are considered.

A transformed rational function method and exact solutions to the 3+1 dimensional Jimbo–Miwa equation

Abstract A direct approach to exact solutions of nonlinear partial differential equations is proposed, by using rational function transformations. The new method provides a more systematical and convenient handling of the solution process of nonlinear equations, unifying the tanh-function type methods, the homogeneous balance method, the exp-function method, the mapping method, and the F -expansion type methods. Its key point is to search for rational solutions to variable-coefficient ordinary differential equations transformed from given partial differential equations. As an application, the construction problem of exact solutions to the 3 + 1 dimensional Jimbo–Miwa equation is treated, together with a Backlund transformation.

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.

Integrable theory of the perturbation equations

Abstract An integrable theory is developed for the perturbation equations engendered from small disturbances of solutions. It includes various integrable properties of the perturbation equations, such as hereditary recursion operators, master symmetries, linear representations (Lax and zero curvature representations) and Hamiltonian structures, and provides us with a method of generating hereditary operators, Hamiltonian operators and symplectic operators starting from the known ones. The resulting perturbation equations give rise to a sort of integrable coupling of soliton equations. Two examples (MKdV hierarchy and KP equation) are carefully carried out.

Complexiton solutions to the Korteweg-de Vries equation

Abstract A novel class of explicit exact solutions to the Korteweg–de Vries equation is presented through its bilinear form. Such solutions possess singularities of combinations of trigonometric function waves and exponential function waves which have different travelling speeds of new type. The functions used in the Wronskian determinants are derived from eigenfunctions of the Schrodinger spectral problem associated with complex eigenvalues, and thus the resulting solutions are called complexiton solutions. Illustrative examples of complexiton solutions are exhibited.

Linear superposition principle applying to Hirota bilinear equations

A linear superposition principle of exponential traveling waves is analyzed for Hirota bilinear equations, with an aim to construct a specific sub-class of N-soliton solutions formed by linear combinations of exponential traveling waves. Applications are made for the 3+1 dimensional KP, Jimbo-Miwa and BKP equations, thereby presenting their particular N-wave solutions. An opposite question is also raised and discussed about generating Hirota bilinear equations possessing the indicated N-wave solutions, and a few illustrative examples are presented, together with an algorithm using weights.

Hamiltonian and quasi-Hamiltonian structures associated with semi-direct sums of Lie algebras

The trace variational identity is generalized to zero curvature equations associated with non-semi-simple Lie algebras or, equivalently, Lie algebras possessing degenerate Killing forms. An application of the resulting generalized variational identity to a class of semi-direct sums of Lie algebras in the AKNS case furnishes Hamiltonian and quasi-Hamiltonian structures of the associated integrable couplings. Three examples of integrable couplings for the AKNS hierarchy are presented: one Hamiltonian and two quasi-Hamiltonian.

Solving the (3 + 1)-dimensional generalized KP and BKP equations by the multiple exp-function algorithm

The multiple exp-function algorithm, as a generalization of Hirota’s perturbation scheme, is used to construct multiple wave solutions to the (3 + 1)-dimensional generalized KP and BKP equations. The resulting solutions involve generic phase shifts and wave frequencies containing many existing choices. It is also pointed out that the presented phase shifts for the two considered equations are all not of Hirota type.

An explicit symmetry constraint for the Lax pairs and the adjoint Lax pairs of AKNS systems

Abstract An explicit symmetry constraint is proposed for the Lax pairs and the adjoint Lax pairs of AKNS systems. The corresponding Lax pairs and adjoint Lax pairs are nonlinearized into a hierarchy of commutative, finite-dimensional integrable Hamiltonian systems in the Liouville sense and thus an involutive representation of solutions of AKNS systems is obtained. The purpose of this Letter is to elucidate that the nonlinearization method (i.e. a kind of symmetry constraint method) of integrable systems can be applied to the Lax pairs and the adjoint Lax pairs associated with integrable systems.