## Abstract

Applying the truncated Painlevé expansion to the (2+1)-dimensional asymmetric Nizhnik–Novikov–Veselov (ANNV) system, some Bäcklund transformations (BTs) including auto BT and non-auto BT are obtained. The auto BT leads to a nonlocal symmetry which corresponds to the residual of the truncated Painlevé expansion and the related nonlocal symmetry group is presented with the help of the localization procedure. Further, it is shown that the ANNV system has a consistent Riccati expansion (CRE). Stemming from the consistent tan-function expansion (CTE), which is a special form of CRE, some complex interaction solutions between soliton and arbitrary other seed waves of the ANNV system are readily constructed, such as bight-dark soliton solution, dark-dark soliton solution, soliton-cnoidal interaction solutions, solitoff solutions and so on.

## 1 Introduction

To find exact solutions of nonlinear systems is a difficult and tedious but very important and meaningful work. One knows that the symmetry method established by Lie and developed by Olver [1] and Bluman and Kumei [2] is very effective for constructing explicit solutions of both integrable and nonintegrable systems. Because the procedure for computing Lie point symmetry is standard and thorough, more and more attention is paid to the nonlocal symmetry which is known to be difficult found and applied. In latter studies, starting from the Darboux transformation (DT) [3–7] and Bäcklund transformation (BT) [8], kinds of nonlocal symmetries are obtained and successfully localized to construct abundant novel solutions, such as the exact interaction solutions amongst solitons and other complicated waves. In Ref. [9], Bluman and Yang introduce a new and complementary method for constructing nonlocally related PDE systems, which is on the basis of each point symmetry.

To study the integrability of nonlinear systems, Painlevé analysis is also one of the best approaches. Recently Lou [10, 11] finds that the residues with respect to the singular manifold of the truncated Painlevé expansions, which are ignored for a long time, may be just the nonlocal symmetries of the original system. The localization of this type of nonlocal symmetry, which is called residual symmetry, seems easily performed than that coming from DT and BT. Moreover, based on the truncated Painlevé expansion, a consistent Riccati expansion (CRE) method [12] is proposed to identify CRE solvable systems. It is found that the CRE method is valid to find exact solutions of nonlinear systems and provide strong signals to clarify possible integrable models. The author [12] points that although not all the integrable systems are CRE solvable, it is strongly indicated that the CRE solvable systems are integrable. It is clear that various integrable systems are CRE solvable [12], such as the Korteweg de-Vries (KdV) equation, the fifth-order KdV equation, the Sawada–Kortera eqquation, the Kaup–Kupershmidt equation, the Kadomtsev–Petviashvili equation, the Boussinesq equation, the AKNS (Ablowitz–Kaup–Newell–Segur) system, the Sine–Gordon equation, the modified asymmetric Veselov–Novikov equation, the dispersive water wave equation and the Burgers equation.

In this article, we concentrate on investigating the residual symmetry and the CRE solvability of the (2+1)-dimensional asymmetric Nizhnik–Novikov–Veselov (ANNV) system [also named the (2+1)-dimensional KdV equation or Boiti–Leon–Manna–Pempinelli equation]:

which was firstly derived by Boiti et al. [13] using the idea of the weak Lax pair and may be considered as a model for an incompressible fluid where *u* and *v* are the components of the (dimensionless) velocity [14]. The spectral transformation for this system has been investigated in Refs. [13, 15]. The nonclassical symmetries, Painlevé property and similarity solutions of the system have been studied by Clarkson and Mansfield [16]. This system has also been considered in [17] as a generalization to (2+1)-dimensions of the results from Hirota and Satsuma [18]. Lou and Hu [19] reobtained this system from the inner parameter-dependent symmetry constraint of the KP equation and found it to be an asymmetric part of the Nizhnik–Novikov–Veselov equation. The conditional similarity reductions are studied in [20] and abundant localized excitations such as dromion solutions [21] and ring solitons [22] are presented by the variable separation approach.

This article is organized as follows. In Section 2, two types of BT and the nonlocal symmetry of the ANNV system are obtained by the Painlevé expansion approach. Then the nonlocal symmetry is localized by introducing another four dependent variables and the corresponding finite group transformation is found. In Section 3, the CRE property and the CTE (which is a special form of CRE) property for the ANNV system are investigated. Based on the CTE method, luxuriant explicit solutions including bight–dark soliton, dark–dark soliton, soliton–cnoidal waves and solitoff solutions are constructed. The last section contains a summary and discussion.

## 2 Nonlocal Symmetry from the Painlevé Expansion

### 2.1 BTs and Nonlocal Symmetry from the Painlevé Expansion

For the (2+1)-dimensional ANNV system (1), there exists a truncated Painlevé expansion

with *u*_{0}, *u*_{1}, *u*_{2}, *v*_{0}, *v*_{1}, *v*_{2}, *ϕ* being the functions of *x*, *y* and *t*.

Substituting (2) into (1), we have

and

By vanishing all the coefficients of different powers of

and the function *ϕ* satisfies

with

Here *C*, *K* and *S* are the usual Schwartzian variables, which keep the Möbious (conformal) invariance property

Due to the aforementioned Möbious invariance, (7) and (8) possess the point symmetries in the form of

with *κ*_{0}, *κ*_{1} and *κ*_{2} being constants. Furthermore, one can verify that (7) and (8) are consistent because *H*_{yyt}=*H*_{yty} is identically satisfied.

On the one hand, from the standard truncated Painlevé expansion (2), we have the following BT theorems of (1).

**Theorem 1(a) (auto-BT theorem)**. If the function *ϕ* satisfies (7) and (8), then

is an auto-BT between the solutions {*u, v*} and {*u*_{0}, *v*_{0}} of the ANNV system (1).

**Theorem 1(b) (non-auto-BT theorem)**. If the function *ϕ* satisfies (7) and (8), then the formula (6) is a non-auto BT between the *ϕ* and the solution {*u*_{0}, *v*_{0}} of the ANNV system (1).

On the other hand, one knows that if *σ*^{u} and *σ*^{ν} are the symmetry of *u* and *v* in (1) respectively, it requires

Comparing the coefficients of *u*_{0} and *v*_{0} satisfy (1), *u*_{1} and *v*_{1} are just the symmetry with respect to *u*_{0} and *v*_{0}, which are called “residual symmetries”. Hence, the ANNV system possesses a symmetry

with

By solving *ϕ* from (16), one knows that the residual symmetries (15) of *u* and *v* are nonlocal.

### 2.2 Localization of the Nonlocal Symmetry

To find out the group transformation

corresponding to the nonlocal symmetry (15), we have to solve the following initial value problem

with ϵ being the infinitesimal parameter.

However, it is difficult to solve (17) about *u̅* and *v̅* due to the intrusion of the function *ϕ* and its differentiations. For this, we need to prolong (1) such that the nonlocal symmetry (15) becomes Lie point symmetry of a larger system. To realize this localization, we introduce four new dependent variables *g*_{1}, *g*_{2}, *h*_{1} and *h*_{2} by

Now one can easily verify that the nonlocal symmetry (15) of the original system (1) becomes a Lie point symmetry of the prolonged system including (1), (16) and (18):

Correspondingly, the initial value problem (17) is changed as

Then the solution of the initial value problem (17) leads to the following group theorem for the enlarged system.

**Theorem 2 (group)**. If {*u*, *v*, *g*_{1}, *g*_{2}, *h*_{1}, *h*_{2}, *ϕ*} is a solution of the prolonged system (1), (16) and (18), so is {*u̅*, *v̅*, *g̅*_{1}, *g̅*_{2}, *h̅*_{1}, *h̅*_{2}, *ϕ̅*} with

Theorem 2 shows us an interesting result that the nonlocal symmetry (15) coming from the truncated Painlevé expansion is just the infinitesimal form of the group (21). Furthermore, we notice that the BT theorem 1(a) and group theorem 2 are equivalent because the singularity manifold equations (1), (16) and (18) are form invariant under the transformation

## 3 CRE Solvable

### 3.1 CRE Solvable

For the ANNV system, we aim to look for the following possible truncated Painlevé expansion solution

where *R*(*w*) is a solution of the Riccati equation

and *r*_{0}, *r*_{1}, *r*_{2} are arbitrary constants. It requires all the expansion coefficient functions *u*_{i} and *v*_{i} be determined by vanishing all the coefficients of the like powers of *R*(*w*) after substituting (22) with (23) into the ANNV system. We write down the final results

with

and the function *w* should satisfy

One can see that (26) and (27) are more complicated than (7) and (8). Luckily (26) and (27) are also consistent because *H*_{1yyt}=*H*_{1yty} is identically satisfied. At this point, we call the (2+1)-dimensional ANNV system be CRE solvable.

### 3.2 CTE solvable and BT

Obviously, the Riccati equation (23) has a special solution

while the truncated expansion solution (22) becomes

where *w* are determined by (24), (26) and (27) with *r*_{0}=1, *r*_{1}=0 and *r*_{2}=– 1.

In view of the consistent tanh-function expansion (CTE) (29), which is a special case of CRE, the ANNV system is called CTE solvable. It is quite clear that a CRE solvable system must be CTE solvable, and vice versa. If a system is CTE solvable, one may directly construct some important explicit solutions, such as the soliton and the interactions between a soliton and a cnoidal wave. To leave this clear, we write down the following non-auto Bäcklund transformation which comes from the aforementioned CTE theorem and use it to find exact solutions.

**Theorem 3 (BT)**. If *w* is a solution of (26) and (27) with *r*_{0}=1, *r*_{1}=0 and *r*_{2}=– 1, then

is a solution of the ANNV system (1), where {*u*_{0}, *v*_{0}} is given by (24) with *δ*=4 and *r*_{1}=0.

### 3.3 Exact solutions from Theorem 3

Obviously, it seems more difficult to solve (26) and (27). However, it is interesting that one can derive some nontrivial solutions of the ANNV system from some quite trivial solutions of (26) and (27) by means of theorem 3. Here are some interesting examples.

**Example 1.***Soliton solutions.* In theorem 3, we take a quite trivial straight-line solution for *w*, saying

with *k, l, d, d*_{0} being arbitrary constants. Substituting (30) with the line solution (31) into (1) yields the following soliton solution

Figure 1 displays a special bright–dark soliton solution for *u* and a dark–dark soliton solution for *v* shown by (32) and (33) at *t*=1 with the parameter selected as

**Example 2.***Soliton-cnoidal waves.* In Ref. [12], it is shown that many CTE solvable systems possess the interaction solutions between solitons and cnoidal periodic waves. Here, to find this type of interaction solutions, let

where *W*_{1} ≡ *W*_{1}(*ξ*)=*W*_{ξ} satisfy

with *a*_{0}, *a*_{1}, *a*_{2}, *a*_{3}, *a*_{4} being constants. Substituting (35) with (36) into theorem 3, it requires

and

Then the system (1) has the explicit solution expressed as

It is clear that (36) has abundant explicit solutions in terms of Jacobi elliptic functions. Hence, the explicit solution (39) exhibits the interactions between a soliton and a cnoidal periodic wave.

To show this soliton+cnoidal wave solutions more intuitively, we just take a a simple solution of (36) as

which leads (39) to a dark soliton residing on a cnoidal wave background, saying

The parameters in (41) and (42) should satisfy

In Figure 2, we plot a dark soliton coupled to a cnoidal wave background expressed by (41) and (42) with

This kind of solution describing solitons moving on a cnoidal wave background instead of on the plane continuous wave background is very important in the real world and can be easily applicable to the analysis of physically interesting processes; see Refs. [4, 5].

**Example 3.***Solitoff solutions*. In theorem 3, if we let

where *P*_{1} ≡ *P*_{1}(*X*)=*P*_{X} satisfy

one can obtain the following explicit solution of the ANNV system (1)

with

Figure 3 shows the structures of the two-solitoff solution and three-solitoff solution for the quantity *u* and *v* shown by (47) and (48), respectively, with

and

## 4 Summary and Discussion

In this article, some Bäcklund transformations, the nonlocal symmetry and different types of exact solutions for the ANNV system are presented. On the one hand, applying the Painlevé expansion method to the ANNV system, two BTs including auto BT and non-auto BT are obtained. It is shown that the residual of the truncated Painlevé expansion in the non-auto BT is just a nonlocal symmetry of the ANNV system, which is also called residual symmetry. Because the residual symmetry is closely related to the Schwartzian variable, we can readily localize it to Lie point symmetries by introducing other auxiliary dependent variables. By enlarging the ANNV system to a prolonged system of eight equations, the corresponding transformation group of the nonlocal symmetry is found, which manifests that the residual symmetry is just the infinitesimal form (or the generator) of the group.

On the other hand, the ANNV system is proved integrable under the mean of consistent Riccati expansion (CRE). As a special form of CRE, the consistent tanh-function expansion (CTE) method for the ANNV system leads to another non-auto BT theorem from which various exact explicit solutions are obtained, such as bight–dark soliton solution, dark-dark soliton solution, soliton–cnoidal interaction solutions, solitoff solutions. The CTE integrability of the ANNV system leaves it clear that the interaction solutions between soliton and arbitrary other seed waves can be constructed simply by plus a straight line solution on a general solution in the non-auto BT.

To find interaction solutions between different types of nonlinear excitations which may display new physical applications is an interesting and meaningful work. With regard to the CRE (or CTE) method, it is worthwhile to detect more associated *w* equations, such as (31), (35) with (36) and (45) with (46). How to combine the truncated Painlevé expansion approach and other methods to construct more residual symmetries and BTs for the ANNV system and other integrable systems, based on which one can obtain more exact solutions, will be further studied in our future researches.

## Acknowledgments

The project is supported by the Zhejiang Provincial Natural Science Foundation of China under Grant No. LQ13A010014, the National Natural Science Foundation of China (Grant Nos. 11326164, 11401528, 11435005, 11375090), Global Change Research Program of China (No.2015CB953904), Research Fund for the Doctoral Program of Higher Education of China (No. 20120076110024), Innovative Research Team Program of the National Natural Science Foundation of China (Grant No. 61321064), Shanghai Knowledge Service Platform for Trustworthy Internet of Things under Grant No. ZF1213.

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**Received:**2015-6-4

**Accepted:**2015-7-9

**Published Online:**2015-8-4

**Published in Print:**2015-9-1

©2015 by De Gruyter