Riemann theta functions periodic wave solutions and rational characteristics for the nonlinear equations

In this paper, based on a multidimensional Riemann theta function, a lucid and straightforward generalization of the Hirota-Riemann method is presented to explicitly construct multiperiodic Riemann theta functions periodic wave solutions for nonlinear equations such as the Caudrey-Dodd-Gibbon-Sawada-Kotera equation and (2+1)-dimensional breaking soliton equation. Among these periodic waves, the one-periodic waves are well-known cnoidal waves, their surface pattern is one-dimensional, and often they are used as one-dimensional models of periodic waves. The two-periodic waves are a direct generalization of one-periodic waves, their surface pattern is two-dimensional so that they have two independent spatial periods in two independent horizontal directions. A limiting procedure is presented to analyze in detail, asymptotic behavior of the multiperiodic waves and the relations between the periodic wave solutions and soliton solutions are rigorously established. This generalized Hirota-Riemann method can also be demonstrated on a class variety of nonlinear difference equations such as Toeplitz lattice equation.     

Riemann theta functions periodic wave solutions and rational characteristics for the (1+1)-dimensional and (2+1)-dimensional Ito equation

In this paper, based on a multidimensional Riemann theta function, a lucid and straightforward way is presented to explicitly construct multiperiodic Riemann theta functions periodic waves solutions for nonlinear differential equation such as the (1+1)-dimensional and (2+1)-dimensional Ito equations. Among these periodic waves, the one-periodic waves are well-known cnoidal waves, their surface pattern is one-dimensional, and often they are used as one-dimensional models of periodic waves. The two-periodic waves are a direct generalization of one-periodic waves, their surface pattern is two dimensional that they have two independent spatial periods in two independent horizontal directions. A limiting procedure is presented to analyze asymptotic behavior of the multiperiodic periodic waves in details and the relations between the periodic wave solutions and soliton solutions are rigorously established.     

Zhiber-Shabat方程的孤立波解与周期波解

结合齐次平衡法原理并利用F展开法,再次研究了Zhiber-Shabat方程的各种椭圆函数周期解.当椭圆函数的模m分别趋于1或0时,利用这些椭圆函数周期解,得到了Zhiber-Shabat方程的各种孤子解和三角函数周期解,从而丰富了相关文献中关于Zhiber-Shabat波方程的解的类型.     

Bifurcation of periodic solutions for a semilinear wave equation

Bifurcation of time periodic solutions and their regularity are proved for a semilinear wave equation, u_tt?u_xx?λu=f(λ,x,u),x?(0,π), t?R, together with Dirichlet or Neumann boundary conditions at x = 0 and x = π. The set of values of the real parameter λ where bifurcation from the trivial solution u = 0 occurs is dense in $\text{R}$.     

New solitary wave solutions and periodic wave solutions for the compound KdV equation

In this paper, for compound KdV equation, four new solitary wave solutions in the form of hyperbolic secant function and six periodic wave solutions in the form of cosine function are obtained by using undetermined coefficient method. On three different layers, the velocity interval which ensures that bell-shaped solitary wave solutions and periodic wave solutions exist synchronously is obtained, respectively. The length of the interval is related to coefficients of the two nonlinear terms.     

Exact periodic wave solutions for the hKdV equation

In this paper, by using the Hirota bilinear method and the Jacobian theta functions for the higher order KdV equation, the existence of periodic wave solutions with one and two period are obtained. The asymptotic properties of the periodic wave solutions are analyzed in detail. It is shown that the well-known soliton solutions can be reduced from the periodic wave solutions.     

A kind of explicit Riemann theta functions periodic waves solutions for discrete soliton equations

In this paper, Riemann theta functions are used to construct one-theta function and two-theta functions solutions to a class of Hirota bilinear equations, such as extended version of the discrete mKdV equation and deautonomization of the two-dimensional Toda lattice equation. To get the Riemann theta function periodic waves solutions (the quasi-periodic solutions), this method is direct and simple which use only the identities of the theta functions.     

Multiple periodic solutions of a superlinear forced wave equation

We study the existence of forced vibrations of nonlinear wave equation: (*) $$\begin{array}{*{20}c} {u_{tt} - u_{xx} + g(u) = f(x,t),} & {(x,t) \in (0,\pi ) \times R,} \\ {\begin{array}{*{20}c} {u(0,t) = u(\pi ,t) = 0,} \\ {u(x,t + 2\pi ) = u(x,t),} \\ \end{array} } & {\begin{array}{*{20}c} {t \in R,} \\ {(x,t) \in (0,\pi ) \times R,} \\ \end{array} } \\ \end{array}$$ whereg(ξ)∈C(R,R)is a function with superlinear growth and f(x, t) is a function which is 2π-periodic in t. Under the suitable growth condition on g(ξ), we prove the existence of infinitely many solution of (*) for any given f(x, t).     

一个组合方程的单孤子解和周期尖波解

构造一个组合方程的单孤子解和周期尖波解.应用格林函数的性质,以及求一个非线性偏微分方程(简称PDE)弱解的方法.求出了这个组合方程的单孤子解和周期尖波解,推广了前人的研究成果.     

New exact homoclinic wave and periodic wave solutions for the Ginzburg-Landau equation

New exact solutions including homoclinic wave and periodic wave solutions for the 2D Ginzburg-Landau equation are obtained using the auxiliary function method and the -expansion method, respectively. The solutions are expressed by the hyperbolic functions and the trigonometric functions. There result shows that there exists a kink wave solution which tends to one and the same periodic wave solution as time tends to infinite.     

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 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.     

Existence of periodic traveling wave solutions for the Ostrovsky equation

We are concerned with the Ostrovsky equation, which is derived from the theory of weakly nonlinear long surface and internal waves in shallow water under the presence of rotation. On the basis of the variational method, we show the existence of periodic traveling wave solutions. Copyright © 2008 John Wiley & Sons, Ltd.     

On theta characteristics of a compact Riemann surface

Let σ be a nontrivial automorphism of a compact connected Riemann surface X of genus at least two. Assume that σ fixes each of the theta characteristics of X. We prove that X is hyperelliptic, and σ is the unique hyperelliptic involution of X.     

New peakon, solitary wave and periodic wave solutions for the modified Camassa-Holm equation

In this paper, an independent variable transformation is introduced to solve the modified Camassa-Holm equation using the bifurcation theory and the method of phase portrait analysis. Some peakons, solitary waves and periodic waves are found and their exact parametric representations in explicit form and in implicit form are obtained.     

Applications of F-expansion to periodic wave solutions for a new Hamiltonian amplitude equation

We present an extended F-expansion method for finding periodic wave solutions of nonlinear evolution equations in mathematical physics, which can be thought of as a concentration of extended Jacobi elliptic function expansion method proposed more recently. By using the F-expansion, without calculating Jacobi elliptic functions, we obtain simultaneously many periodic wave solutions expressed by various Jacobi elliptic functions for the new Hamiltonian amplitude equation introduced by Wadati et al. When the modulus m approaches to 1 and 0, then the hyperbolic function solutions (including the solitary wave solutions) and trigonometric function solutions are also given respectively. As the parameter ε goes to zero, the new Hamiltonian amplitude equation becomes the well-known nonlinear Schrödinger equation (NLS), and at least there are 37 kinds of solutions of NLS can be derived from the solutions of the new Hamiltonian amplitude equation.