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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.

In this paper, we consider the Jaulent-Miodek (JM) Equation [

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 [

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-inelastic- interaction 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).

We consisder the isospectral problem introduced in [

and the auxiliary spectral problem

From the zero curvature equation

We introduce a transformation

with

The Lax pair (2.1) and (2.2) is transformed into a new Lax pair

and

We suppose that

where

Let

and (2.2). From (2.3), there exist constants

with

There are

The unknown

From (2.8) and (2.9), we have

which means

Proposition 1. Let

Through the transformation (2.3) with (2.4), the isospectral problem (2.1) is transformed into (2.6) with

where

Proof. Let

It is easy to see that

Then all

where

and

By comparing the coefficients of

From (2.21), (2.23) and (2.25), together with (2.11), (2.13), (2.14), (2.19), (2.20) and (2.24), we respectively get

Comparing with (2.4) and (2.18), we find that

Remark. When

Proposition 2. Let

where

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.

Since the transformation (2.3) with (2.14) transforms the Lax pair (2.1) and (2.2) into the same Lax pair (2.6) and (2.7), the transformation

In this section, by using of the above obtained DT, we get new solutions of the JM Equation (1.1).

For simplicity, taking

with

According to (2.10), we get

In the following, we discuss the two cases

1) For

with

with

As

2) For

where

with

The exact solution of the JM Equation (1.1) is

When the parameters are suitably chosen, the solution (3.8) describes the elastic-inelastic-interaction coexistence phenomenon, i.e. the elastic and fission interactions coexist at the same time (see

In

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