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