B¨acklund Transformation for Solving a (3+1)-Dimensional Integrable Equation

1. Introduction

Searching for Bäcklund transformations (BTS) of nonlinear differential equations has been an important topic in the field of soliton theory and integrable systems. The methods for investigating Bäcklund transformations of (1+1)-dimensional differential equations have widely studied [1,2,3,4]. However, for the Bäcklund transformations of high-dimensional integrable equations, only some special approaches were presented. For example, in Ref [5], the Bäcklund transformation connected with the general two-dimensional Gelfand-Dikij-Zakharov-Shabat spectral problems was found. In Refs

In this paper, we study the generalized (3+1)-dimensional Kortweg-de Vries (3dKdV,that is 3dKP) equation starting from a set of vector fields which can reduce to the celebrated KP equation which describes disturbances in a weakly dispersive, weakly nonlinear medium. Again the 3dKP equation obtained in the paper also reduces to the standard three-dimensional KP equation (see [14])

$$u_t+u_{xxx}+6u{u}_x+3{\sigma }^2{\partial }_x^{-1}(u_{yy}+u_{zz})=0,$$

(1.1)

which has an important role in fluid dynamics as well as in fields such as plasma physics. In [15], solitary wave and elliptic functional solutions of the 3dKP equation (1.1) have been studied. And soliton-like solutions and period-form solutions of the equation have been constructed [16]. Where u depends on time variable t and space variables $x,y$, ${\sigma }^2=\pm 1$. At the same time, we need know that ${\partial }_x^{-1}$ is an inverse operator of ${\partial }_x$, ${\partial }_x^{-1}{\partial }_x={\partial }_x{\partial }_x^{-1}=1$. If ${\partial }_z=0$, equation (1.1) reduces to the (2+1)-dimensional KP equation [17].

This paper will be organized as follows: Utilizing the Lax pair transformation, the Bäcklund transformation and a type of new soliton solutions are constructed in Section 2 and Section 3, respectively. In Section 4, a superposition formula is obtained via the Bäcklund transformation.

2. A Bäcklund Transformation of Three Dimensions

Consider the three-dimensional vector fields addtoresetequationsection

$$\begin{array}{cc}\hfill L_1=& {\partial }_x^2+\sigma {\partial }_y+\sigma {\partial }_z+u,\hfill \\ \hfill L_2=& {\partial }_t+4{\partial }_x^3+6u{\partial }_x+3u_x-3\sigma {\partial }_x^{-1}{u}_y-3\delta {\partial }_x^{-1}{u}_z+\gamma {\partial }_x,\hfill \end{array}$$

(2.1)

where $\sigma$, $\delta$ and $\gamma$ are constants independent of x, y, z and t. It is easy calculate the commutativity condition of the Lax pair

$$\begin{array}{cc}\hfill L_1\psi =& \alpha \psi ,\hfill \\ \hfill L_2\psi =& \beta \psi ,\hfill \end{array}$$

(2.2)

implies that

$$\begin{array}{c}\hfill u_{xt}+u_{xxxx}+\gamma u_{xx}+3{\left(u^2\right)}_{xx}+3{\sigma }^2{u}_{yy}+6\sigma \delta u_{yz}+3{\delta }^2{u}_{zz}=0,\end{array}$$

(2.3)

where $\alpha$, $\beta$ are eigenvalues independent of $\overrightarrow{x}=(x,y,z,t)$. When $\gamma =0$, $\sigma =1$, $\delta =0$ equation (2.3) reduces to well-known KP equation

$$\begin{array}{c}\hfill u_{xt}+u_{xxxx}+3{\left(u^2\right)}_{xx}+3u_{yy}=0.\end{array}$$

(2.4)

When $\gamma =0$, $\sigma =\pm i$, $\delta =0$ equation (2.3) becomes

$$u_{xt}+u_{xxxx}+3{\left(u^2\right)}_{xx}-3u_{yy}=0.$$

(2.5)

The positive sign refers to negative dispersion in (2.4), while the negative sign refers to the positive dispersion in (2.5)

Let $\varphi =ln\psi$, $u=v_x$, we can obtain that equation (2.2) has the following form

$$\begin{array}{cc}\hfill & {\varphi }_{xx}+{\varphi }_x^2+\sigma {\varphi }_y+\sigma {\varphi }_z+v_x=\alpha ,\hfill \\ \hfill & {\varphi }_t+4({\varphi }_{xxx}+3{\varphi }_x{\varphi }_{xx}+{\varphi }_x^3)+6v_x{\varphi }_x+3v_{xx}-3\sigma (v_y+v_z)+\gamma {\varphi }_x=\beta .\hfill \end{array}$$

(2.6)

Eliminating v in equations (2.6), we get the following nonlinear equation

$$\begin{array}{cc}\hfill & {\varphi }_t+{\varphi }_{xxx}-2{\varphi }_x^3+6\alpha {\varphi }_x+6\sigma \delta {\int }^x{\varphi }_{yz}dx+3{\sigma }^2{\int }^x{\varphi }_{yy}dx\hfill \\ \hfill & +3{\sigma }^2{\int }^x{\varphi }_{zz}dx-6\sigma {\int }^x{\varphi }_y{\varphi }_{xx}dx-6\sigma {\int }^x{\varphi }_z{\varphi }_{xx}dx+\gamma {\varphi }_x=\beta .\hfill \end{array}$$

(2.7)

We observe that for every $(\varphi ,\beta ,\sigma ,\delta ,\gamma )$ which satisfy equation (2.7), the $(-\varphi ,-\beta ,-\sigma ,-\delta ,\gamma )$ satisfy the equation as well. For this new sequence $(-\varphi ,-\beta ,-\sigma ,-\delta ,\gamma )$, there is a corresponding solution $u^{\prime }\equiv v_x^{\prime }$ of equation (2.3) such that

$$\begin{array}{cc}\hfill & -{\varphi }_{xx}+{\varphi }_x^2+\sigma {\varphi }_y+\sigma {\varphi }_z+v_x^{\prime }=\alpha ,\hfill \\ \hfill & -{\varphi }_t+4(-{\varphi }_{xxx}+3{\varphi }_x{\varphi }_{xx}-{\varphi }_x^3)-6v_x^{\prime }{\varphi }_x+3v_{xx}^{\prime }-3\sigma (v_y^{\prime }+v_z^{\prime })-\gamma {\varphi }_x=-\beta .\hfill \end{array}$$

(2.8)

Taking the difference and sum of equation (2.6) and equation (2.8), one gets that

$$\begin{array}{cc}\hfill & 2{\varphi }_{xx}+v_x-v_x^{\prime }=0,\hfill \\ \hfill & 2{\varphi }_t+8{\varphi }_{xxx}+8{\varphi }_x^3+6{(v+v^{\prime })}_x{\varphi }_x+3{(v-v^{\prime })}_{xx}\hfill \\ \hfill & -3\sigma {(v^{\prime }-v)}_y+3\sigma {(v^{\prime }-v)}_z+2\gamma {\varphi }_x=2\beta ,\hfill \end{array}$$

(2.9)

and

$$\begin{array}{cc}\hfill & 2{\varphi }_x^2+2\sigma {\varphi }_y+2\sigma {\varphi }_z+v_x+v_x^{\prime }=2\alpha ,\hfill \\ \hfill & 24{\varphi }_x{\varphi }_{xx}+6{(v-v^{\prime })}_x{\varphi }_x+3{(v-v^{\prime })}_{xx}-3\sigma {(v+v^{\prime })}_y-3\sigma {(v+v^{\prime })}_z=0.\hfill \end{array}$$

(2.10)

From the first equation in (2.9), we have

$$\varphi ={\displaystyle \frac{1}{2}}{\int }^x(v-v^{\prime })dx.$$

(2.11)

Inserting equation (2.11) into equations (2.9) and (2.10) yields the Bäcklund transformation

$$\begin{array}{cc}\hfill & {(v^{\prime }-v)}^2+2{\int }^x{(v-v^{\prime })}_{y}dx+2{\int }^x{(v-v^{\prime })}_{z}dx+2{(v+v^{\prime })}_x=0,\hfill \\ \hfill & 24{\varphi }_x{\varphi }_{xx}+6{(v-v^{\prime })}_x{\varphi }_x+3{(v-v^{\prime })}_{xx}-3\sigma {(v+v^{\prime })}_y-3\sigma {(v+v^{\prime })}_z=0.\hfill \end{array}$$

(2.12)

When taking $v=0$ in equation (2.1), it becomes

$$\begin{array}{cc}\hfill & {\psi }_{xx}+\sigma {\psi }_y+\sigma {\psi }_z=0,\hfill \\ \hfill & {\psi }_t+4{\psi }_{xxx}+\gamma {\psi }_x=0.\hfill \end{array}$$

(2.13)

A solution to (2.13) can be taken as

$$\begin{array}{cc}\hfill \psi & =\sum _l{\Lambda }_l{e}^{-lx-{\displaystyle \frac{l^2}{2\delta }}y-{\displaystyle \frac{l^2}{2\delta }}z+(4l^3+\gamma l)t}\hfill \\ \hfill & \equiv \sum _l{\Lambda }_l{e}^{{\theta }_l}.\hfill \end{array}$$

(2.14)

The summation runs over all complex values of l and ${\Lambda }_l$ is a spectral function. A new solution is given by

$$\begin{array}{cc}\hfill u^{\prime }& =v_x^{\prime }=2{\varphi }_{xx}=2{(ln\varphi )}_{xx}\hfill \\ \hfill & =2{\displaystyle \frac{(-{\displaystyle \sum _l}{\Lambda }_l{l}^2{e}^{{\theta }_l})\left({\displaystyle \sum _l}{\Lambda }_l{e}^{{\theta }_l}\right)-{\left({\displaystyle \sum _l}l{\Lambda }_l{e}^{{\theta }_l}\right)}^2}{{\left({\displaystyle \sum _l}{\Lambda }_l{e}^{{\theta }_l}\right)}^2}},\hfill \end{array}$$

(2.15)

where

$${\theta }_i=-l_{i}x-{\displaystyle \frac{l_i^2}{2\delta }}y-{\displaystyle \frac{l_i^2}{2\delta }}z+(4l_i^3+\gamma l_i)t,i=1,2.$$

(2.16)

Special choice of ${\Lambda }_l$ will result in special solutions, then we consider two different cases. When ${\Lambda }_l={\delta }_{l,l_1}+{\delta }_{l,l_2}$, from (2.15) one infers that

$$u^{\prime }={\displaystyle \frac{1}{2}}{(l_1-l_2)}^2{sech}^2{\displaystyle \frac{1}{2}}[(l_2-l_1)x+{\displaystyle \frac{l_2^2-l_1^2}{2\sigma }}y+{\displaystyle \frac{l_2^2-l_1^2}{2\delta }}z+(4(l_1^3-l_2^3+\gamma (l_1-l_2))t],$$

(2.17)

which is a three-dimensional soliton solution with amplitude ${\displaystyle \frac{1}{2}}{(l_1-l_2)}^2$ and velocity

$$P_x=:-4(l_1^2+l_2^2+l_1{l}_2+\gamma ),P_y=:-{\displaystyle \frac{8\sigma (l_1^2+l_2^2+l_1{l}_2+\gamma )}{l_1+l_2}},P_z=:-{\displaystyle \frac{8\delta (l_1^2+l_2^2+l_1{l}_2+\gamma )}{l_1+l_2}}.$$

(2.18)

When ${\Lambda }_l={\delta }_{l,l_1}-{\delta }_{l,l_2}$, from (2.15) we have

$$u^{\prime }={\displaystyle \frac{1}{2}}{(l_1-l_2)}^{2}csc{h}^2{\displaystyle \frac{1}{2}}[(l_2-l_1)x+{\displaystyle \frac{l_2^2-l_1^2}{2\sigma }}y+{\displaystyle \frac{l_2^2-l_1^2}{2\delta }}z+\left(4(l_1^3-l_2^3+\gamma (l_1-l_2))t\right],$$

(2.19)

which is a singular solution.

3. A Type of New Soliton Solutions

The spectral problem (2.13) not only possesses the soliton solutions such as (2.17) and (2.19), but also does the following solution addtoresetequationsection

$$\psi =\sum _l{\displaystyle \frac{{\Lambda }_l{e}^{{\theta }_l}}{1+\Delta e^{{\theta }_l}}},$$

(3.1)

where $\Delta$ is a constant. When ${\Lambda }_l={\delta }_{l,l_1}+{\delta }_{l,l_2}$, it is easy to find

$$\begin{array}{cc}\hfill & \psi =P+Q,\hfill \\ \hfill & {\psi }_x=l_1\Delta P^2+l_2\Delta Q^2-l_{1}P-l_{2}Q,\hfill \\ \hfill & {\psi }_{xx}=2l_1^2{\Delta }^2{P}^3+2l_2^2\Delta Q^3-3l_1^2\Delta P^2-3l_2^2\Delta Q^2+l_1^{2}P+l_2^{2}Q,\hfill \end{array}$$

(3.2)

where

$$P={\displaystyle \frac{{\Lambda }_l{e}^{{\theta }_1}}{1+\Delta e^{{\theta }_1}}},Q={\displaystyle \frac{{\Lambda }_l{e}^{{\theta }_2}}{1+\Delta e^{{\theta }_2}}}.$$

(3.3)

Therefore, a new solution to equation (2.1) is given by

$$u^{\prime }=2[{\displaystyle \frac{l_1^{2}P-3l_1^2\Delta P^2+2l_1^2{\Delta }^2{P}^3+l_2^{2}Q-3l_2^2\Delta Q^2+2l_2^2\Delta Q^3}{P+Q}}-{\left({\displaystyle \frac{P-Q+l_1\Delta P^2+l_2\Delta Q^2)}{P+Q}}\right)}^2].$$

(3.4)

When ${\Lambda }_l={\delta }_{l,l_1}+{\delta }_{l,l_2}$, equation (3.1) becomes

$$\psi ={\displaystyle \frac{{\Lambda }_l{e}^{{\theta }_1}}{1+\Delta e^{{\theta }_1}}}-{\displaystyle \frac{{\Lambda }_l{e}^{{\theta }_2}}{1+\Delta e^{{\theta }_2}}}=P-Q.$$

(3.5)

Similar to the above calculation, we obtain another new solution to equation (2.1)

$$u^{\prime }=2[{\displaystyle \frac{l_1^{2}P-3l_1^2\Delta P^2+2l_1^2{\Delta }^2{P}^3-l_2^{2}Q+3l_2^2\Delta Q^2+2l_2^2\Delta Q^3}{P-Q}}-{\left({\displaystyle \frac{-l_{1}P+l_{2}Q+l_1\Delta P^2-l_2\Delta Q^2)}{P-Q}}\right)}^2].$$

(3.6)

4. Superposition Formula

The advantage of the Bäcklund transformation is that a superposition of solutions can be derived. This superposition allows us to construct more complex solutions only using algebraic methods. To get the superposition formula, let $v_1$ be a solution given by Bäcklund transformation from a known solution $v_0$ with some spectral function ${\Lambda }_{1,l}$, $v_2$ be a second solution generated from $v_0$ with spectral function ${\Lambda }_{2,l}$, $v_3$ a third solution obtained from $v_1$ with spectral functions ${\Lambda }_{2,l}$. According to the definition and Bäcklund transformation (2.12), one gets that addtoresetequationsection

$$\begin{array}{cc}\hfill & {(v_1-v_0)}^2+2{\int }^x{(v_1-v_0)}_{y}dx+2{\int }^x{(v_1-v_0)}_{z}dx+2{(v_1+v_0)}_x=0,\hfill \\ \hfill & {(v_2-v_0)}^2+2{\int }^x{(v_2-v_0)}_{y}dx+2{\int }^x{(v_2-v_0)}_{z}dx+2{(v_2+v_0)}_x=0,\hfill \\ \hfill & {(v_3-v_1)}^2+2{\int }^x{(v_3-v_1)}_{y}dx+2{\int }^x{(v_3-v_1)}_{z}dx+2{(v_3+v_1)}_x=0.\hfill \end{array}$$

(4.1)

Form (4.1) we can deduce that

$$v_3+v_0-2(ln|v_1-v_2{\left|\right)}_x=v_1+v_2,$$

(4.2)

which is just right the superposition formula. Starting form $v_0=0$, we have $v_1$ and $v_2$ as given in Eq. (2.17) and (2.19) with spectral functions ${\Lambda }_{1,l}$ and ${\Lambda }_{2,l}$, respectively, which can be referred to as single-spectrum solutions. Substituting them into formula (4.2), we can get a solution $v_3$ which contains two spectral function ${\Lambda }_{1,l}$ and ${\Lambda }_{2,l}$. Going on in terms of the calculation, we can get many new solutions to Eq. (2.3).

5. Conclusions

In this paper, the Bäcklund transformation (2.12) of 3dKP was constructed by means of the Lax pair (2.2), which in turn yielded several types of soliton solutions and superposition formulas for the equation. Extending the low-dimensional equations to high-dimensional equations and investigating their various properties and solutions, such as multi-solitons, multi-breathers, rational solutions etc., via Bäcklund transformation method may be one of the future research topics. Various types of exact solutions of the KP equation (1.1) in three spatial dimensions were being analyzed. The methods and results presented in this paper may provide a good inspiration for dealing with similar high-dimensional nonlinear equations [18,19,20,21,22].