By the function transformation and the first integral of the ordinary differential equations, the problem of solving the solutions of the double sine-Gordon equation and the treble sine-Gordon equation is researched, and the new solutions are obtained. First, the problem of solving the solutions of the double sine-Gordon equation and the treble sine-Gordon equation is changed to the problem of solving the solutions of the nonlinear ordinary differential equation. Second, with the help of the B?cklund transformation and the nonlinear superposition formula of solutions of the first kind of elliptic equation and the Riccati equation, the new infinite sequence soliton-like solutions of two kinds of sine-Gordon equations are constructed.
Refs. [
where p and
Refs. [
where
In this paper, by the function transformation and the first integral of the ordinary differential equations, the problem of solving the solutions of the double sine-Gordon equation and the treble sine-Gordon equation is changed to the problem of solving the solutions of the nonlinear ordinary differential equation. Based on these, with the help of the Bäcklund transformation and the nonlinear superposition formula of solutions of the first kind of elliptic equation and the Riccati equation, the new infinite sequence soliton-like solutions of two kinds of sine-Gordon equations are constructed, which are consisting of Riemann
The relative conclusions of the Bäcklund transformation of some kinds of ordinary differential equations introduced as follows are very important in constructing the new solutions of the two kinds of sine-Gordon equations.
Theorem 2.1 When
According to the relative conclusions of the Riccati Equation [
Theorem 2.2 There is the following fitting Bäcklund transformation between the ordinary differential Equa- tion (3) and the first kind of elliptic Equation (6).
Then we put forward the fitting Bäcklund transformation between the ordinary differential Equation (3) and the first kind of elliptic Equation (6) in some cases.
Case 1. When
Case 2. When
Case 3. When
Case 4. When
Here
Theorem 2.3 If
Here
Theorem 2.4 The first kind of elliptic Equation (6) has the following some kinds of solutions.
Case 1. The Riemann
When
When
When
where
Case 2. The Jacobi elliptic function new solutions of the first kind of elliptic equation
According to the periodicity of the Jacobi elliptic function, some kinds of new solutions of the first kind of elliptic equation are obtained, some new solutions [
When
When
When
where
Case 3. The other new solutions of the first kind of elliptic equation
When
Substituting the functional transformation
By the functional transformation, the ordinary differential Equation (33) is changed to the ordinary differential Equations (34)
Then by the functional transformation (35), the ordinary differential Equations (34) is changed to the ordinary differential Equations (36)
We can obtain the first integral of the ordinary differential Equations (36) as follows
where
Substituting the first integral (37) into the first equation of the ordinary differential Equations (36) yields the following ordinary differential equation
where
With the help of the relative conclusions of some kinds of ordinary differential equations introduced in Part 2, the new infinite sequence solutions of the treble sine-Gordon equations are constructed
When
where
Case 1. The new infinite sequence smooth-type soliton-like solutions
When the coefficients of the ordinary differential Equation (39)
where
kind of elliptic Equation (6), and
If the
If the
If
Case 2. The new infinite sequence peak-type soliton solutions
When
where
kind of elliptic Equation (15), and
If the
If the
The ordinary differential Equations (38) is changed to the following the ordinary differential equation when
where
Equation (42) is changed to the Riccati Equation (44) with the help of the following functional transformation
By the following superposition formula, the new infinite sequence soliton-like solutions of the treble sine-Gordon equations are obtained, which are consisting of hyperbolic function, trigonometric function and rational function.
where p and r are arbitrary constants that are not all zero.
When
where
where
Substituting the functional transformation
By the functional transformation, the ordinary differential Equation (48) is changed to the following two ordinary differential equations
The two ordinary differential equations have the following first integral
where
Substituting the first integral into the first equation of the ordinary differential Equations (49) and (50) severally yields the following two ordinary differential equation
where
When
where
By the method to construct the new infinite sequence solutions of the treble sine-Gordon equation, we can also obtain the new infinite sequence solutions of the double sine-Gordon equation (not given here).
By the auxiliary equation method, many kinds of smooth type soliton, tense type soliton and peak soliton and so on new solutions of the nonlinear evolution equations have been obtained [
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. NJZY16180) and the Natural Science Foundation of Inner Mongolia Autonomous Region, China (Grant No. 2015MS0128).
Yu Mei Bai,Taogetusang , (2016) The New Infinite Sequence Solutions of Multiple Sine-Gordon Equations. Journal of Applied Mathematics and Physics,04,796-805. doi: 10.4236/jamp.2016.44090