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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.

In Refs. [

where

where

In Refs. [

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 func- tion 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 two, substituting the first integrals (21), (22) separately into the first equation of ordinary differential Equations (19), (20) yields two first order nonlinear ordinary differential Equations (23) and (24). Substituting the first integrals (33), (34) separately into the first equation of ordinary differential Equations (31), (32) yields two first order nonlinear ordinary differential Equations (35) and (36).

Step three, using function transformation, two first order nonlinear ordinary differential Equations (23) and (24) (or (23) and (24)) can be 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 [

Here

Assume the solutions of a class of nonlinear evolutionary Equation (10) as

where

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

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

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

The ordinary differential Equations (25) and (26) can be expressed by the following forms.

And by the following transformation (29) and (30), the two ordinary differential Equations (27), (28) can be expressed by the form of (31), (32).

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

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.

When

where

In (38)-(43),

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.

Case 5. The new infinite sequence complexion solutions consisting of the trigonometric function and the rational function of a class of nonlinear evolutionary Equation (10).

Substituting the solution determined by the superposition formula (39), (43) into (37) yields the new infinite sequence complexion solutions consisting of the trigonometric function and the rational function.

Case 6. The rational function type new infinite sequence solutions of a class of nonlinear evolutionary Equation (10).

Substituting the solution determined by the superposition formula (40) and (43) into (37) yields the rational function type new infinite sequence solutions.

When

With the help of the relative conclusions of the second kind of elliptic equation [

Case 1. When

Case 2. When

Case 3. When

In Refs. [

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

When the coefficients of nonlinear evolutionary Equations (1) are

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 con- structed.

Project supported by the Natural Science Foundation of China (Grant No. 11361040), the Science Research Foundation of Institution of Higher Education of Inner Mongolia Autonomous Region, China (Grant No. NJZY12031) and the Natural Science Foundation of Inner Mongolia Autonomous Region, China (Grant No. 2015MS0128).

LinaYi,JundongBao, Taogetusang, (2015) The New Infinite Sequence Complexion Solutions of a Kind of Nonlinear Evolutionary Equations. Journal of Applied Mathematics and Physics,03,1624-1632. doi: 10.4236/jamp.2015.312187