Abstract:
We generalized the Jacobi elliptic function expansion method and obtained some new exact periodic solutions of Zakharov equations, thus replenished the known results of the equation using this method.

Abstract:
We put a direct new method to construct the rational Jacobi elliptic solutions for nonlinear differential difference equations which may be called the rational Jacobi elliptic functions method. We use the rational Jacobi elliptic function method to construct many new exact solutions for some nonlinear differential difference equations in mathematical physics via the lattice equation and the discrete nonlinear Schrodinger equation with a saturable nonlinearity. The proposed method is more effective and powerful to obtain the exact solutions for nonlinear differential difference equations.

Abstract:
This paper is concerned with the system of Zakharov equations which involves the interactions between Langmuir and ion-acoustic waves in plasma. Abundant explicit and exact solutions of the system of Zakharov equations are derived uniformly by using the first integral method. These exact solutions are include that of the solitary wave solutions of bell-type for n and E, the solitary wave solutions of kink-type for E and bell-type for n, the singular traveling wave solutions, periodic wave solutions of triangle functions, Jacobi elliptic function doubly periodic solutions, and Weierstrass elliptic function doubly periodic wave solutions. The results obtained confirm that the first integral method is an efficient technique for analytic treatment of a wide variety of nonlinear systems of partial differential equations.

Abstract:
With the help of the generalized Jacobi elliptic function, an improved Jacobi elliptic function method is used to construct exact traveling wave solutions of the nonlinear partial differential equations in a unified way. A class of nonlinear Schrödinger-type equations including the generalized Zakharov system, the Rangwala-Rao equation, and the Chen-Lee-Lin equation are investigated, and the exact solutions are derived with the aid of the homogenous balance principle.

Abstract:
In this work, Jacobi elliptic function solutions for integrable nonlinear equations using F-expansion method are represented. KdV and Boussinesq equations are considered and new results are obtained.Key Words: Jacobi elliptic functions; F-expansion method; Solitary waves; Periodic solutions

Abstract:
In this work, Jacobi elliptic function solutions for integrable nonlinear equations using F-expansion method are represented. KdV and Boussinesq equations are considered and new results are obtained. Key Words: Jacobi elliptic functions; F-expansion method; Solitary waves; Periodic solutions

Abstract:
Some doubly-periodic solutions of the Zakharov--Kuznetsov equation are presented. Our approach is to introduce an auxiliary ordinary differential equation and use its Jacobi elliptic function solutions to construct doubly-periodic solutions of the Zakharov--Kuznetsov equation, which has been derived by Gottwald as a two-dimensional model for nonlinear Rossby waves. When the modulus k \rightarrow 1, these solutions reduce to the solitary wave solutions of the equation.

Abstract:
Jacobi elliptic function expansion method is applied to construct the exact envelope periodic solutions of nonlinear wave equations. These envelope periodic solutions obtained by this method include the envelope shock wave solutions or the envelope solitary wave solutions.

Abstract:
In the paper, to construct new infinite sequence exact solutions of nonlinear evolution equations, several kinds of new solutions of the second kind of elliptic equation B cklund transformation are proposed. The KdV equation containing variable coefficients and forcible term, combined with (2+1)-dimensional and (3+1)-dimensional Zakharov-Kuznetsov equation with variable coefficients is taken as example to construct new infinite sequence exact solutions of these equations with the help of symbolic computation system Mathematica, which include infinite sequence compact soliton solutions of Jacobi elliptic function and triangular function, and infinite sequence peak soliton solutions.

Abstract:
From any given Frobenius manifold one may construct a so-called dual structure which, while not satisfying the full axioms of a Frobenius manifold, shares many of its essential features, such as the existence of a prepotential satisfying the WDVV equations of associativity. Jacobi group orbit spaces naturally carry the structures of a Frobenius manifold and hence there exists a dual prepotential. In this paper this dual prepotential is constructed and expressed in terms of the elliptic polylogarithm function of Beilinson and Levin.