^{1}

^{*}

^{2}

^{1}

A (2 + 1) dimensional KdV-mKdV equation is proposed and integrability in the sense of Painlevé and some exact solutions are discussed. The Bäcklund transformation and bilinear equations are obtained through Painlevé analysis. Some exact solutions are deduced by Hirota method and generalized Wronskian method.

Recently high dimensional nonlinear partial differential or difference equations attract much interest. Both inte- grable and non-integrable equations have their physical and mathematical values but the former posses some special properties such as infinite conservation laws and symmetries, multi-soliton solutions, Bäcklund and Darboux transformation (c.f. [

where subscript means a partial derivative such as

that if

to treat the integral appearing in equation. The Equation (1) is then rewritten as

We will prove it has Painlevé property firstly, then deduce a Bäcklund transformation and bilinear equation. Using bilinear equation we can construct Wronskian solutions and present some exact solutions finally.

Painlevé analysis method is an important method for testing integrability [

Step 1. Expanding the solution of a PDE as Laurent series of a singular manifold

where

Step 2. If all exponents

Step 3. If all resonances are integers, check the resonance conditions in each Laurent expansion.

Conclusion. If no obstruction is found in Steps 1 - 3 for every dominant balances, then the Painlevé test is satisfied.

The situation of high dimensional case is similar. For step 1, we can simply let

Substituting them into (2, 3) gives us

where

Insert them into (2, 3) and equal coefficients of both side of

From them we work out

To get resonances we collect the coefficient of

where F, G are functions of

Now we proceed to verify the resonance conditions. First we consider

or equivalently

The part of

and it is true by employing

where

In a similar way, collecting the terms of

where

we need to verify

because

we can complete the proof of compatible condition. It is a turn to consider

where

and

where

Its resonance condition is verified similarly but is more complex. Thus we prove that (2 + 1) dimensional KdV-mKdV equation passes Painlevé test.

Now we consider to truncate the series (6). To meet this end we must let

and combine the equation satisfied by

Furthermore, If we continue to set

and

The condition

Using (20)-(23) we may truncate the series. Thus we indeed get a Bäcklund transformation by noting (22, 23). But it is more important pointing that the identities (20)-(23) have only two independent expressions, say (22, 23). Applying the definition of Schwartzian derivative

we simplify them as a concise form, i.e. so called Schwartzian derivative equation

It is the condition satisfied by function

In this section we will give the bilinear equation of Equation (1) and present some exact solutions from it. The truncation form (19) suggests us to try the transformation

We first take an integral with respect to x on Equation (1). Then eliminate the remaining integral operator by setting

where D is bilinear operator. Thus we can transfer Equation (1) into

Equations (26, 27) are bilinear equations of (1). To find its solutions we set ^{*} means complex conjugation. Expanding f as perturbation series

and substituting it into bilinear equations, equaling coefficients of power of

Take

where

The coefficient of

If we take

then after substituting it into (29, 30) we know relations

are valid. Again compare coefficient of

When employing (34),

are obtained. After that we consider coefficient of

The r.h.s is computed to zero. Thus we may truncate the perturbation series and 2-soliton solution is got as

Further, keeping these results in mind we can conjecture the N-soliton solution taking on

where

Wronskian technique is one of the powerful methods in finding exact solutions of nonlinear integrable evolution equation [

and

where

Supposing that vectors

where A is a non-singular real constant

To get down to our work we need the help of two Lemmas, we list out them first.

Lemma 1 ( [

Lemma 2 ( [

We first treat bilinear Equations (26). Computing derivatives of Wronskians

When apply Lemma 2 into Wronskians

Then adding it to (44) gives us

which equals zero by using Lemma 1. Now we can focus our attention on the bilinear Equation (27). We also calculate the derivative of Wronskians

Then

Again using Lemma 2, we produce two identities as follows:

The substitution of (48, 49) into (47) yields

To vanish r.h.s of this equation we apply Lemma 1 again, which give us a valuable identity

Multiply

It is because of

In a same way, we deduce

Thus we complete the proof that

Now we present some exact solutions as examples. Firstly, we may write out the expression of spectral vector

where

where I is

and

then spectral vector

The solutions given by (25) are solitons solutions in this situation. In fact, when

This gives the same solution as (41) or simplified form:

which is a two-soliton solution. We can also take into account other solutions. For instance, let

Then we find

Taking

The correspondent solution of Equation (1) is

or simplified form

This is known as a complexiton solution (c.f. [

Utilizing Painlevé test we prove the integrability of a (2 + 1) dimensional KdV-mKdV equation in the sense of Painlevé. And in the mean time a Bäcklund transformation is produced. Through bilinear equation we get several exact solutions by Hirota method and generalized Wronskian method. Some explicit formulas of exact solutions are obtained. Particularly, 2-soliton solution and complexiton solutions are presented as examples.

The authors are grateful to editors and referees for their very careful works. In the mean time, the authors thank to the referees for giving helpful advices.

This work is partly supported by Chinese National Social Science Foundation (Grant Number: CNSSF: 13CJY037) Research on the indemnificatory Apartment Construction Based on Residential Integration.