Abstract
In this letter, we consider a new two-component system. By its reciprocal relation with the first negative flow in the AKNS hierarchy, we construct three Darboux-Bäcklund transformations for the new system and obtain some soliton-like solutions.
1. Introduction
It is well known that Darboux transformation (DT) and Bäcklund transformation (BT) play important roles in establishing new exact solutions of integrable systems from old ones [1–4]. For example, it may be used to construct multisoliton solutions from a trivial seed. For some classical integrable systems, one may construct DT/BT via the gauge transformation. But for others such as the Camassa-Holm equation and the complex short pulse equation, one can not construct their DTs/BTs directly, because they involve the independent variable . Fortunately, the reciprocal transformation provides a useful tool [5–8]. There are many other methods to study exact solutions of integrable systems, see Refs. [9–13].
Recently, while discussing reciprocal transformations of negative flows of some classical integrable hierarchies, Li and Wu [14] proposed some new systems; one of them reads
A Lax pair of (1) has been derived by using its reciprocal relation with the first negative flow in the AKNS hierarchy. Based on this Lax pair and using the standard approach, one may present infinitely many conserved quantities for it. However, its DT/BT and exact solutions are still unknown.
In this letter, we will construct three BTs and provide some soliton-like solutions for (1). The paper is arranged as follows. In Section 2, we will construct three BTs of (1) by using its reciprocal link with the first negative flow in the AKNS hierarchy. In Section 3, we will construct the multisoliton solutions of (1) by using the BTs discovered above.
2. Darboux Transformations for the Negative AKNS System
As pointed out in [14], the system (1) admits the following Lax pair
with The system (1) possesses a conservation law of the form , which allows a reciprocal transformation
Through this reciprocal transformation, it is easy to show that (1) is changed to the first negative flow in the AKNS hierarchy and the Lax pair (2) is converted to
Now, we turn our attention to the negative AKNS system [15–17]. The famous DTs for the negative AKNS system may be summarized as the following proposition.
Proposition 1. The Lax pair (5) is covariant with respect to the following three DTs: (1)The first one is given bywhere is a special solution of the Lax pair (5) at . (2)The second one readswhere is a special solution of the Lax pair (5) at . (3)The third one is provided bywhere , , and is a special solution of the Lax presentation (5) at .
3. DTs and Some Soliton-Like Solutions of the New System
In this section, we will use the three DTs for the negative AKNS system in Proposition 1 to construct three BTs, which involve the independent variable , for the new system (1) [8]. The key point of the procedure is to determine the coordinate transformation between and via the inverse transformation of (3).
Firstly, let us apply the first DT (6) to the Lax pair (5), which leads to the BT between and . In particular, we have which leads to
Considering the inverse of (3), we may obtain a BT of system (2) as where and is an arbitrary constant.
Hereafter, as an example of use of this BT, let us choose a seed of (4) as where is an arbitrary constant. It follows immediately that with Now, substituting them into (11) yields
A profile of 2-kink like solution is plotted in Figures 1 and 2 at
Secondly, applying the DT (7) to the Lax presentation (5) yields which leads to
Using the inverse of (3), we may obtain the second BT of system (2) as with and is an arbitrary constant. Choosing the same seed as (12), one may obtain
The profile is similar to the first case, and hence, we omit it here.
Finally, imposing the DT (8) to the Lax pair (5) and setting leads to where
Furthermore, a straightforward calculation shows that
Combining (19) and (21), we arrive at
Hence, we acquire the third BT of system (2), which is where and is an arbitrary constant. Choosing the same initial solution as (12) gives rise to
where Substituting them into (23), we get
A profile of 2-kink like solution is plotted in Figures 3 and 4 at
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work is partially supported by the National Natural Science Foundation of China (11805071, 11747010, and 11871232), the Fujian Province Science Foundation for Youths (2019J05092), and the Fundamental Research Funds for the Central Universities (ZQN-803).