N-Fold Darboux Transformation of the Jaulent-Miodek Equation

In this paper, based on the Lax pair of the Jaulent-Miodek spectral problem, we construct the Darboux transformation of the Jaulent-Miodek Equation. Then from a trivial solution, we get the exact solutions of the Jaulent-Miodek Equation. We obtain a kink-type soliton and a bell-kink-type soliton. Particularly, we obtain the exact solutions which describe the elastic-inelastic-interaction coexistence phenomenon.


Introduction
In this paper, we consider the Jaulent-Miodek (JM) Equation [1] 3 2 0, x xxx q qq r r q r qr q We study the exact solutions of the JM Equation (1.1) by using Darboux transformation (DT), which is an effective method to get exact solutions from the trivial solutions of the nonlinear partial differential equations based on the Lax pairs [2]- [11].As to the higher JM Equation, authors used several methods considering the travellling wave solutions [12]- [14].For the solutions of the JM Equation (1.1), in [1], the solitary wave solutions have been obtained by Darboux transformation.In this paper, we start from a different Lax pair to get some new exact solutions.
This paper is arranged as follows.Based on the Lax pair of the JM Equation (1.1), in Section 2, we deduce a basic DT of the JM Equation (1.1).In Section 3, from a trivial solution, we get solitary wave solutions of the JM Equation (1.1).Particularly, we obtain the bell-kink-type solitary wave solutions.We also get the elastic-inelasticinteraction coexistence phenomenon for the JM Equation (1.1).To the author's best knowledge, this is a new phenomenon for the JM Equation (1.1).
To prove Proposition 2, we need to use Proposition 1 and the JM Equation (1.1), together with the help of the mathematical software (such as Mathematica).Although the idea of the proof for Proposition 2 is the same as Proposition 1, it is much more tedious and is omitted for brevity.

Exact Solutions
In this section, by using of the above obtained DT, we get new solutions of the JM Equation (1.1).
For simplicity, taking 0 q r = = , we get two basic solutions of the Lax pair (2.1) and (2. Sinh In the following, we discuss the two cases 1 N = and 2 N = . 1) For 1 N = , from (2.9) and (2.11 ), we have ( ) ( ) ( ) .This solution is similar with the solution in [11].As 1 1 r > − , this is a solitary wave solution where [ ] is a bell-kink-type soliton, i.e. soliton is composed of a bell-type wave and a kink-type wave (see Figure 1).
In Figure 3 soliton are head-on interactions (this is an elastic interaction), K1 kink-type soliton, K3 kink-type soliton and K5 kink-type soliton fuse into K135 kink-type soliton (this is a inelastic interaction).The solution [ ] 2 r is a solitary wave solution, which is the same as [ ] 2 q , but the solitons are the bell-kink-type (see also Figure 3).This phenomenon has been described in the Whitham-Broer-Kaup shallow-water-wave model [16].It seems to be new for the JM Equation.

Let
solutions of the Lax pair(2.1) exact solution of the JM Equation (1.1) is [ ]