求Nizhnik-Novikov-Veselov方程新精确解

引入改进的F-广义方法,并将其应用于（2＋1）维Nizhnik-Novikov-Veselov（NNV）方程.在符号计算软件的帮助下,可以得到NNV方程的许多新解.该方法用于获取包括雅可比椭圆函数解的一系列解,在数学物理中可应用于其他的非线性偏微分方程.     

Global small solutions to the critical radial Dirac equation with potential

We solve globally a radial cubic Dirac equation perturbed with a small potential, with data of small critical norm H¹. The main tool is a new endpoint estimate of the perturbed Dirac flow for a class of radial-type initial data.     

New exact solutions to the generalized Burgers-Huxley equation

Based on He’s Exp-function method, a series of new exact solutions of the generalized Burger-Huxley equation have been obtained. It is shown that the Exp-function method is straightforward and concise, and its applications are promising.     

New exact solutions to the nonautonomous Liouville equation

We obtain new formulas for the exact analytic solutions to the nonautonomous elliptic Liouville equation in the two-dimensional coordinate space with the free function dependent specially on an arbitrary harmonic function. We present new exact solutions to the wave Liouville equation with two arbitrary functions, providing original formulas for the general solution for the classical (autonomous) and wave Liouville equations. Some equivalence transformations are presented for the elliptic Liouville equation depending on conjugate harmonic functions. In particular, we indicate a transformation that reduces the equation under study to an autonomous form.     

New exact solutions of a nonlinear integrable equation

Analytic solutions of the partial differential equations are needed to explain many phenomena seen in thermodynamics, aerodynamics, plasma physics, and other fields. In this paper, variational principle is analyzed of the integrable nonlinear Korteweg–de Vries (KdV) typed equation. In addition, exact solutions of this equation are obtained by using various methods such as direct integration, homogeneous balance method, Exp-function method, and Kudryashov method.     

New exact periodic wave solutions for the Dullin-Gottwald-Holm equation

In this paper, the Dullin-Gottwald-Holm equation is studied using semi-inverse method and integral bifurcation method. New periodic waves such as peakon-like periodic wave, compacton-like periodic wave and singular periodic wave are found and their dynamical behaviors and certain strange phenomena are explained using the proposed criterion. The exact parametric representations of these waves are also presented.     

New exact solutions for the nonlinear wave equation with fifth-order nonlinear term

The purpose of this paper is to reveal the dynamical behavior of the nonlinear wave equation with fifth-order nonlinear term, and provides its bounded traveling wave solutions. Applying the bifurcation theory of planar dynamical systems, we depict phase portraits of the traveling wave system corresponding to this equation under various parameter conditions. Through discussing the bifurcation of phase portraits, we obtain all explicit expressions of solitary wave solutions and kink wave solutions. Further, we investigate the relation between the bounded orbit of the traveling wave system and the energy level h. By analyzing the energy level constant h, we get all possible periodic wave solutions.     

Approximate bound state solutions of Dirac equation with Hulthén potential including Coulomb-like tensor potential

We solve the Dirac equation approximately for the attractive scalar S(r) and repulsive vector V(r) Hulthén potentials including a Coulomb-like tensor potential with arbitrary spin-orbit coupling quantum number κ. In the framework of the spin and pseudospin symmetric concept, we obtain the analytic energy spectrum and the corresponding two-component upper- and lower-spinors of the two Dirac particles by means of the Nikiforov-Uvarov method in closed form. The limit of zero tensor coupling and the non-relativistic solution are obtained. The energy spectrum for various levels is presented for several κ values under the condition of exact spin symmetry in the presence or absence of tensor coupling.     

New exact solutions for the Kawahara equation using Exp-function method

In this paper, we applied the Exp-function method to solve the Kawahara equation. This method can be used to obtain new exact solutions and periodic solutions with parameters are obtained. It is shown that the Exp-function method, with the help of symbolic computation, provides a very effective and powerful (mathematical tools) for discrete nonlinear evolution equations in mathematical physics.     

New exact solutions to the Fitzhugh–Nagumo equation

By the first integral method, a series of new exact solutions of the Fitzhugh–Nagumo equation have been obtained. It is shown that this method is one of the most effective approaches to obtain the exact solutions of the nonlinear evolution equations, especially for nonintegrable models.     

New exact periodic solitary wave solutions for Kadomtsev-Petviashvili equation

Exact periodic solitary wave solutions for Kadomtsev-Petviashvili equation are obtained by using the Hirota bilinear method. The result shows that there exists periodic solitary waves in the different directions for (2 + 1)-dimensional Kadomtsev-Petviashvili equation.     

New exact solutions for a generalized variable-coefficient KdV equation

New exact solutions for a generalized variable-coefficient KdV equation were obtained using the generalized expansion method [R. Sabry, M.A. Zahran, E.G. Fan, Phys. Lett. A 326 (2004) 93]. The obtained solutions include solitary wave solutions besides Jacobi and Weierstrass doubly periodic wave solutions.     

New exact general solutions of Abel equation of the second kind

The Abel equation of the second kind

[g₀(x)+g₁(x)u]u^′=f₀(x)+f₁(x)u+f₂(x)u²     

Symmetry reduced and new exact non-traveling wave solutions of potential Kadomtsev-Petviashvili equation with p-power

With the aid of Maple symbolic computation and Lie group method, PKPp equation is reduced to some (1 + 1)-dimensional partial differential equations, in which there are linear PDE with constant coefficients, nonlinear PDE with constant coefficients, and nonlinear PDE with variable coefficients. Using the separation of variables, homoclinic test technique and auxiliary equation methods, we obtain new abundant exact non-traveling solution with arbitrary functions for the PKPp.     

New exact solutions to the -dimensional Konopelchenko–Dubrovsky equation

In this paper, the extended tanh method, the sech–csch ansatz, the Hirota’s bilinear formalism combined with the simplified Hereman form and the Darboux transformation method are applied to determine the traveling wave solutions and other kinds of exact solutions for the (2+1)-dimensional Konopelchenko–Dubrovsky equation and abundant new soliton solutions, kink solutions, periodic wave solutions and complexiton solutions are formally derived. The work confirms the significant features of the employed methods and shows the variety of the obtained solutions.