New Exact Solutions of the (2+1)-dimensional Ginzburg-landau Equation

A novel identical reforming of differential equation and the high order auxiliary methods are used to construct solitary solutions and periodic solutions of (2 + 1)-Dimensional Ginzburg-Landau equation. It is shown that the high order auxiliary method, with the help of symbolic computation, provides a powerful mathematical tool for solving nonlinear equations arising in mathematical physics.


INTRODUCTION
The investigation of exact traveling wave solutions to nonlinear evolution equations plays an important role in the study of nonlinear physical phenomena.The wave phenomena are observed in fluid dynamics, plasma, elastic media, optical fibers, etc. Complex Ginzburg-Landau equation (CGLE) is a major subject in nonlinear optics, which describes the propagation of optical pulses in optic fibers.
The solutions of CGLE have been extensively studied in various aspects since it was derived [1][2][3][4].A new model was introduced by Sakaguchi and Malomed to describe a nonlinear planer waveguide incorporated into a closed optical cavity, a 2D cubic-quintic Ginzburg-Landau equation(CQGLE) with an anisotropy of a novel type which is diffractive in one direction, and diffusive in the other,.some interesting phenomena of this equation at the zero-dispersion point were demonstrated using systematic simulation [5].However, so far it is rarely for seeking the exact solitary wave solutions of this equation in addition to the reference [6].The goal of the present work is searching for exact solutions of the cubic-quintic Ginzburg-Landau equation.
Over the last few decades, directly searching for exact solutions of nonlinear partial differential equations (NPDEs) has become more attractive topic in physical science and nonlinear science.With the rapid development of nonlinear science based on computer algebraic system like Maple package, some new powerful solving methods have been developed, such as multi-wave method [7] homogeneous balance principle [8][9][10], F-expansion method [11][12], extended auxiliary equation method [13][14], and so on.
In this work, exact solutions of the CQGLE are considered.The novel identical where  <0 is a real constant, z and x are the propagation and transverse coordinates, respectively.
is the so-called reduced time, where t is the physical time, and 0 V is the group velocity of the carrier wave.Since u is a complex function, we can assume that Eq. ( 1) have solutions in the form ( , ) ( , , ) ( , where k and ( , ) x  is real to be determined , ( , ) x  is a real unknown function.
Substituting (2) into Eq.(1)and separating the real part and the imaginary part of result yield where are all real constants to be determined.
Using( 11)again we get Eq.( 13) is rewritten as Eq.( 8)can be written as Equation ( 14) and equation ( 15) must be the same equation, comparing their coefficients, we get , 2 Taking value of k and 1 h as above, then substitute them into Eq. ( 14 and (15), we can obtain one and the same equation.

USING HIGHER ORDER AUXILIARY EQUATION FOR SOLVING THE EQUATION
In this section, the CQGL equation is solved by using a higher order auxiliary equation method.We seek for the solutions of Eq. (17) in the form 0 ( ) ( ) in which ( 0,1, 2,..., ) Substituting ( 18) into (17) along with (19), balancing the highest order derivative term with the highest power nonlinear term in Eq. ( 17), we find 1 n  .Therefore, the solution of Eq. ( 17) is the form as follows 01 ( ) ( ) where 01 , AA are constants to be determined.Substituting Eq. (20) into Eq.( 17) to get a polynomial with respect to () F  .Equating to zero the coefficients of all powers of () F  yields a set of algebraic equations for , , , , l A A r r , ( 0, 2, 4,6)  .Solving these equations with Maple, we get the following results Substituting these result into Eq.(20)，we can get the general form solutions of Eq. ( 1) along with Eq. ( 2 )

. Solutions to the CQGL equation
With the solutions of Eq. (19), the exact solutions of the Eq.(1) are obtained as follows