The New Infinite Sequence Complexion Solutions of a Kind of Nonlinear Evolutionary Equations

The method combining the function transformation with the auxiliary equation is presented and the new infinite sequence complexion solutions of a class of nonlinear evolutionary equations are constructed. Step one, according to two function transformations, a class of nonlinear evolutionary equations is changed into two kinds of ordinary differential equations. Step two, using the first integral of the ordinary differential equations, two first order nonlinear ordinary differential equations are obtained. Step three, using function transformation, two first order nonlinear ordinary differential equations are changed to the ordinary differential equation that could be integrated. Step four, the new solutions, Bäcklund transformation and the nonlinear superposition formula of solutions of the ordinary differential equation that could be integrated are applied to construct the new infinite sequence complexion solutions of a class of nonlinear evolutionary equations. These solutions are consisting of two-soliton solutions, two-period solutions and solutions composed of soliton solutions and period solutions.


Introduction
In Refs.[1]- [5], some methods were applied to research the following two nonlinear coupling systems, and a finite number of one soliton solutions were obtained.L. N. Yi et al.
where , h l and m are constants; ρ and σ are scalars.In Refs.[6]- [9], the new two-soliton solutions of the following a class of nonlinear evolutionary equations are constructed.

(
) ( ) ( ) ( ) ( ) ( ) . 6 2 ( ) ( ) . 6 6 . 6 6 In this paper, the method combining the function transformation with the auxiliary equation is applied to construct the new infinite sequence complexion solutions of a class of nonlinear evolutionary Equation (10) by the following steps.
Step one, in Part 2.1, according to function transformation (11), a class of nonlinear evolutionary Equation (10) is changed into two kinds of ordinary differential Equations (13), (14).And then with the help of the function transformation, ordinary differential Equations ( 13), ( 14) can be changed to ordinary differential Equations (19), (20).And the first integrals (21), (22) of the two ordinary differential equations are obtained.In Part 2.2, according to function transformation (12), a class of nonlinear evolutionary Equations (10) is changed into two kinds of ordinary differential Equations (25), (26).And then with the help of the function transformation, ordinary differential Equations ( 25), (26) can be changed to ordinary differential Equations (31), (32).And the first integrals (33), (34) of the two ordinary differential equations are obtained.
Step four, the new solutions, Bäcklund transformation and the nonlinear superposition formula of solutions of the ordinary differential equation that could be integrated [10]- [12] are applied to construct the new infinite sequence complexion solutions consisting of the Riemann θ function, the Jacobi elliptic function, the hyper- bolic function, the trigonometric function and the rational function of a class of nonlinear evolutionary equations (10).These solutions are consisting of two-soliton solutions, two-period solutions and solutions composed of soliton solutions and period solutions.

The Method Combining the Function Transformation with the Auxiliary Equation
Assume the solutions of a class of nonlinear evolutionary Equation (10) as . 2 where , , , , and j ω are constants to be determined, and ,

The First Kind of Function Transformation with the Two Kinds of Ordinary Differential Equations
Substituting the first kind of function transformation (11) into a class of nonlinear evolutionary Equation (10) yields the following conclusions.Case 1.When the coefficients of a class of nonlinear evolutionary equations satisfy the following conditions, the complexion solutions exist.When a class of nonlinear evolutionary Equation ( 10) has the complexion solutions, and the solutions are determined by the following nonlinear ordinary differential equations. ( 3 Case 2. The first integral of the second order nonlinear ordinary differential equations.By the function transformation, the nonlinear ordinary differential Equations ( 13) and ( 14) are changed into the following two ordinary differential equations ( ) ( ) ) And by the following transformation (17) and ( 18), the two ordinary differential Equations ( 15) and ( 16) can be expressed by the form of ( 19) and (20).
( ) By calculating, the following first integral of the ordinary differential Equations ( 19) and (20) are obtained ( ) ( ) Substituting the first integral ( 21) and ( 22) severally into the first equation of the ordinary differential Equations ( 19) and (20) yields the following two ordinary differential equations ( ) where

The Second Kind of Function Transformation with the Two Kinds of Ordinary Differential Equations
Substituting the second kind of function transformation (12) into a class of nonlinear evolutionary Equation (10) yields the following conclusions.Case 1.When the coefficients of a class of nonlinear evolutionary equations satisfy the following conditions, the complexion solutions exist.When a class of nonlinear evolutionary Equation ( 10) has the complexion solutions, and the solutions are determined by the following nonlinear ordinary differential equations.
The ordinary differential Equations ( 25) and ( 26) can be expressed by the following forms.
( ) Case 2. The first integral of the second order nonlinear ordinary differential equations.By calculating, the following first integral of the ordinary differential Equations (31), (32) are obtained.
( ) Substituting the first integral (33) and (34) severally into the first equation of the ordinary differential Equations (31) and (32) yields the following two nonlinear ordinary differential equations where ( )

The New Infinite Sequence Complexion Solutions of a Class of Nonlinear Evolutionary Equations
In some cases, according to the ordinary differential Equations ( 23) and (24) (or (35) and ( 36)), the new infinite sequence complexion solutions of a class of nonlinear evolutionary Equation (10) are constructed.

The New Infinite Sequence Complexion Solutions Consisting of the Hyperbolic Function, the Trigonometric Function and the Rational Function
) where ( ) ( ) 1 m P ξ and ( ) ( ) 1 n Q η are determined by the following superposition formula [10]- [13]. ) In ( 38)-(43), 0 0 , L M and 0 N are arbitrary constants not all equal to zero.Case 1.The hyperbolic function type new infinite sequence complexion two-soliton solutions of a class of nonlinear evolutionary Equation (10).
Substituting the solution determined by the superposition formula (38), (41) into (37) yields the hyperbolic function type new infinite sequence complexion two-soliton solutions.
Case 2. The new infinite sequence complexion solutions consisting of the hyperbolic function and the trigonometric function of a class of nonlinear evolutionary Equation (10).
Substituting the solution determined by the superposition formula (38), ( 42) into (37) yields the new infinite sequence complexion solutions composed of soliton solutions and the period solutions.
Case 3. The new infinite sequence complexion solutions consisting of the hyperbolic function and the rational function of a class of nonlinear evolutionary Equation (10).
Substituting the solution determined by the superposition formula (38) and ( 43) into (37) yields the new infinite sequence complexion solutions consisting of the hyperbolic function and the rational function.
Case 4. The trigonometric function type new infinite sequence two-period solutions of a class of nonlinear evolutionary Equation (10).
Substituting the solution determined by the superposition formula (39), (42) into (37) yields the trigonometric function type new infinite sequence two-period solutions.

Conclusions
In Refs.
[1]- [5], a finite number of one function type one soliton new solutions consisting of the Jacobi elliptic function, the hyperbolic function, the trigonometric function and the rational function of the nonlinear evolutionary Equations ( 1)-(4) were constructed.In Refs.[6]- [9], a finite number of complexion function type twosoliton new solutions consisting of the Jacobi elliptic function, the hyperbolic function, the trigonometric function and the rational function of the nonlinear evolutionary Equation (5) were constructed.In this paper, the method combining the function transformation with the auxiliary equation is presented, and the problem of solving the solutions of a class of nonlinear evolutionary Equation (10) concluding the nonlinear evolutionary Equations ( 1)-( 9) is considered; the new infinite sequence complexion solutions consisting of the Riemann θ function, the Jacobi elliptic function, the hyperbolic function, the trigonometric function and the rational function are obtained.These solutions are consisting of two-soliton solutions, two-period solutions and solutions composed of soliton solutions and period solutions.
When the coefficients of nonlinear evolutionary Equations (1) are 3 that the new infinite sequence complexion solutions exist.So, according to the method given in the Part 3, the new infinite sequence complexion two-soliton and two-periods and so on solutions of nonlinear evolutionary Equation (1) are constructed.
When the coefficients of a class of nonlinear evolutionary Equation ( 10) are the condition that the new infinite sequence complexion solutions exist.Then according to the conclusions given in Part 3 in the paper, the new infinite sequence complexion two-soliton solutions, two-period solutions and solutions composed of soliton solutions and period solutions of nonlinear evolutionary Equation ( 5) are constructed. 1625 and e are constants; ρ and σ are scalars.
the following superposition formula, the new infinite sequence complexion solutions consisting of the hyperbolic function, the trigonometric function and the rational function of a class of nonlinear evolutionary Equation (10) are obtained.

(
new infinite sequence complexion solutions consisting of the Riemann θ function, the Jacobi elliptic function, the hyperbolic function, the trigonometric function and the rational function of a class of nonlinear evolutionary Equation (10) are obtained.Case 2new infinite sequence complexion solutions consisting of the Riemann θ function, the Jacobi elliptic function, the hyperbolic function, the trigonometric function and the rational function of a class of nonlinear evolutionary Equation (10) are obtained.Case 3new infinite sequence complexion solutions consisting of the Riemann θ function, the Jacobi elliptic function, the hyperbolic function, the trigonometric function and the rational function of a class of nonlinear evolutionary Equation (10) are obtained.