With the help of the method that combines the first kind of elliptic equation with the function transformation, some kinds of new composite solutions of a kind of coupled Schrödinger equation are constructed. First, a kind of function transformation is presented, and then the problem of solving solutions of a kind of coupled Schrödinger equation can be changed to the problem of solving solutions of the first kind of elliptic equation. Then, with the help of the conclusions of the Bäcklund transformation and so on of the first kind of elliptic equation, the new infinite sequence composite solutions of a kind of coupled Schrödinger equation are constructed. These solutions are consisting of two-soliton solutions and two-period solutions and so on.
In many researches of the physical problems such as the high frequency movement of plasma, nonlinear optical, nonlinear dissipative system and fluid mechanics and so on, the Schrödinger type equations always appear. Many methods to solving solutions of these nonlinear evolution equations are presented [
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First, a kind of function transformation is presented, and then the problem of solving solutions of a kind of coupled Schrödinger equation can be changed to the problem of solving solutions of the first kind of elliptic equation. Then, with the help of the conclusions of the Bäcklund transformation and so on of the first kind of elliptic equation, the new infinite sequence composite solutions of a kind of coupled Schrödinger equation are constructed. These solutions are new composite solutions consisting of two-soliton solutions, two-period solu- tions and the solutions composed of soliton solutions and period solutions composed in pairs by Riemann
Then we put forward the Bäcklund transformation and so on new conclusions of the first kind of elliptic equation [
Here a, b and c are constants.
Case 1. The Riemann
When
Here
Case 2. The Jacobi elliptic function type new solutions of the first kind of elliptic equation
According to the periodic of Jacobi elliptic function, many kinds of new solutions of the first kind of elliptic equation can be obtained, here we list some kinds of new solutions [
When
Here
Case 3. The other new solutions of the first kind of elliptic equation
When
If
Here a, b and c are the coefficients of first kind of elliptic Equation (5). l is an arbitrary constant not equal to zero.
If
Here a, b and are c the coefficients of first kind of elliptic Equation (5).
When
By the following transformation, Equation (15) can be changed to Riccati equation (17).
Then we put forward the relative conclusions of Riccati equation [
Riccati Equation (18) has the following normal solutions.
Here
If
Here
By the following function transformation (26) (27), the problem of solving solutions of a kind of coupled Schrödinger Equation (3) (4) can be changed to the problem of solving solutions of two first kind of elliptic equation.
Here
When
transformation (26) and (27) into a kind of coupled Schrödinger Equation (3) (4) yields the following nonlinear ordinary differential equations.
The Equations (28) and (29) integrate once then we obtain
Here
By the following superposition formula we obtain the new infinite sequence composite solutions of a kind of coupled Schrödinger Equation (3) (4).
Here
With the help of the relative conclusions the paper part two and part three put forward, we obtain the new infinite sequence solutions of the first kind of elliptic Equation (30) (31). Substituting these solutions separately into Formula (32) (33) yields the new infinite sequence composite solutions of a kind of coupled Schrödinger equation. These solutions are consisting of new solutions composed in pairs by Riemann
When
When
Case 1. The new composite two-period solutions composed by two Riemann
Substituting the solutions obtained by superposition Formula (34) (36) together into Formula (32) (33) yields the new infinite sequence composite two-period solutions composed by two Riemann
Case 2. The new composite two-period solutions composed by Riemann
Substituting the solutions obtained by superposition Formula (34) (37) (or (35) (36)) together into Formula (32) (33) yields the new infinite sequence composite two-period solutions composed by Riemann
Case 3. The new composite two-period solutions composed by two Jacobi elliptic functions.
Substituting the solutions obtained by superposition Formula (35) (37) together into Formula (32) (33) yields the new infinite sequence composite two-period solutions composed by two Jacobi elliptic functions of a kind of coupled Schrödinger equation.
When d0 and d1 are not all equal to zero, construct the new infinite sequence composite solutions.
When
Case 1. The new infinite sequence composite solutions composed by Riemann
Case 2. The new infinite sequence composite solutions composed by Jacobi elliptic function period solution and exponential function soliton solution.
Case 3. The new infinite sequence composite two-soliton solutions composed by two exponential functions.
Case 4. The new infinite sequence composite solutions composed by exponential function type soliton solution and trigonometric function period solution.
Case 5. The new infinite sequence composite two-period solutions composed by Riemann
Case 6. The new infinite sequence composite two-period solutions composed by Jacobi elliptic function period solution and trigonometric function period solution.
Case 7. The new infinite sequence composite two- period solutions composed by two trigonometric functions.
When
Substituting the solutions obtained by the following superposition formula into Formula (32) (33) yields the new infinite sequence composite solutions of a kind of coupled Schrödinger equation.
Here
Case 1. The new infinite sequence composite two-soliton solutions composed by two exponential functions.
Substituting the solutions obtained by superposition formula (38),(40) together into formula (32),(33) yields the new infinite sequence composite two-soliton solutions composed by two exponential functions of a kind of coupled Schrödinger equation.
Case 2. The new infinite sequence composite solutions composed by exponential function type soliton solution and trigonometric function period solution.
Substituting the solutions obtained by superposition Formula (38) (41) (or (39),(40)) together into Formula (32) (33) yields the new infinite sequence composite solutions composed by exponential function type soliton solution and trigonometric function period solution of a kind of coupled Schrödinger equation.
Case 3. The new infinite sequence composite two-period solutions composed by two trigonometric functions.
Substituting the solutions obtained by superposition Formula (39) (41) together into Formula (32) (33) yields the new infinite sequence composite two-period solutions composed by two trigonometric functions of a kind of coupled Schrödinger equation.
Constructing the multiple-soliton solution of nonlinear evolution equation is a very important research of soliton theory. Auxiliary equation method has obtained many achievements in soliton theory. Such as: Literature [
Based on the achievements the auxiliary equation method has obtained, the paper constructs many kinds of new infinite sequence composite solutions of a kind of coupled Schrödinger Equation (3) (4). These solutions are new infinite sequence composite solutions composed in pairs by Riemann
When
meet the condition that
that have been already obtained of a kind of coupled Schrödinger Equation (3) (4), we can construct the new infinite sequence composite solutions of Schrödinger Equation (1) (2).
Project supported by the Natural 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).
Yili Na,Baojun Dong, Taogetusang, (2015) Some Kinds of New Composite Solutions of a Kind of Coupled Schrödinger Equation. Journal of Applied Mathematics and Physics,03,1376-1385. doi: 10.4236/jamp.2015.311165