Bäcklund Transformation for Solving a (3+1)-Dimensional Integrable Equation

Abstract

A new generalized (3+1)-dimensional Kadomtsev–Petviashvil (3dKP) equation is derived from the inverse scattering transform method. This equation can be reduced to the standard KP equation and the well-know (3+1)-dimensional equation. In making use of the Lax pair transformation, a Bäcklund transformation of the generalized (3+1)-dimensional KP equation is constructed and some soliton solutions are produced. Finally, a superposition formula is singled out as well by making use of the Bäcklund transformation. As far as we know, the work presented in this paper has not been studied up to now.

1. Introduction

Searching for Bäcklund transformations (BTS) of nonlinear differential equations has been an important topic in the fields of soliton theory and integrable systems. Methods for investigating the Bäcklund transformation of (1+1)-dimensional differential equations have been widely studied [1,2,3,4]. However, for the Bäcklund transformations of high-dimensional integrable equations, only some special approaches have been presented. For example, in Ref. [5], the Bäcklund transformation connected with the general two-dimensional Gelfand–Dikij–Zakharov–Shabat spectral problems was found. In Ref. [6], the authors studied the well-known KP equation and nonlinear evolution equation starting from a two-dimensional vector field and further obtained the Bäcklund transformation of the KP equation and nonlinear evolution equation. Konopelchenko generalized the AKNS technique to the two-dimensional arbitrary-order matrix spectral problem in [7]. Bäcklund transformations in (2+1) dimensions were found as well. For example, the two-dimensional Korteweg–de Vries-like equation [8], sine-Gordon system [9], and Sawada–Kotera equation [10] were exhibited. It has been found that many high-dimensional equations have a wide range of applications in mathematical physics, including neural network algorithms and various architectures, as well as science and engineering, fiber optics, and fluid mechanics. In recent years, the exact solutions and properties for high-dimensional equations have become a focus of research, especially for equations with more than two spatial variables. Constructing Bäcklund transformations and exact solutions of nonlinear evolution equations in three dimensions has been an important open problem in the field of integrability research. For example, Yin et al. used the Hirota bilinear method to construct a Bäcklund transformation of the (3+1)-dimensional nonlinear evolution equation, which consists of four equations and involves six free parameters [11]. Ma et al. proposed a bilinear Bäcklund transform for the (3+1)-dimensional generalized KP equation and computed two classes of exponential and rational traveling wave solutions with arbitrary wave numbers [12]. We know that soliton solutions have a wide and important role in nonlinear wave theory, elementary particle theory, and other fields, which can help us to understand the algebraic structure and basic properties of 3dKP equations in theory and explain some related natural phenomena in practical applications.

In this paper, we study the (3+1)-dimensional Kortweg–de Vries (3dKdV) equation starting from a set of vector fields that can be reduced to the celebrated equation that describes disturbances in a weakly dispersive, weakly nonlinear medium. Again the 3dKdV equation obtained in this paper can also be reduced to the standard three-dimensional KP equation (see [13])

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

(1)

which has an important role in fluid dynamics as well as in fields such as plasma physics. Here, u depends on time variable t and space variables $x,y$, ${\sigma }^2=\pm 1$. At the same time, we need to 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) reduces to the two-dimensional KP equation [14].

We know that the different types of solutions and properties of the 3dKP equation have been investigated by many experts and scholars. For example, in [15], solitary wave and elliptic functional solutions, and so on, of 3dKP Equation (1) were studied. And soliton-like solutions and period-form solutions of the equation have been constructed [16]. Zhang et al. constructed the M-rogue wave solutions of the 3dKP equation in [17]. In [18], bilinear form, multiple-soliton, breather, and lump solutions were obtained using the Hirota bilinear method. And Muslum et al. examined the single- and singular-soliton properties of the (3+1)-dimensional KP-B equation using the modified extended tanh function and Kudryashov method [19]. In contrast to previous research work, we constructed a Bäcklund transformation of the equation by means of the Lax pair, which in turn yields new soliton solutions of the 3dKP equation, and thus, the equation’s nonlinear dynamical behavior is better studied. The above significant results provide good inspiration for dealing with similar high-dimensional nonlinear equations.

This paper will be organized as follows: With the utilization of 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

In this section, we construct a kind of soliton solution to the 3dKP equation by means of the Bäcklund transformation, which is of great significance for the further study of the exact solutions and properties of the 3dKP equation. Then, we study a Bäcklund transformation of three dimensions. Consider the three-dimensional vector fields

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

(2)

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

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

(3)

imply that

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

(4)

where $\alpha$ and $\beta$ are eigenvalues independent of $\overrightarrow{x}=(x,y,z,t)$. When $\gamma =0$, $\sigma =1$, and $\delta =0$, Equation (4) is reduced to the well-known KP equation

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

(5)

When $\gamma =0$, $\sigma =\pm i$, and $\delta =0$, Equation (4) becomes

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

(6)

The positive sign refers to negative dispersion in (5), while the negative sign refers to the positive dispersion in (6).

Let $\varphi =ln\psi$ and $u=v_x$. We can obtain that Equation (3) has the following form:

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

(7)

Eliminating v in Equation (7), we obtain the following nonlinear equation:

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

(8)

We observe that for every $(\varphi ,\beta ,\sigma ,\delta ,\gamma )$ that satisfy Equation (8), $(-\varphi ,-\beta ,-\sigma ,-\delta ,\gamma )$ satisfy the equation. For this new sequence $(-\varphi ,-\beta ,-\sigma ,-\delta ,\gamma )$, there is a corresponding solution $u^{\prime }\equiv v_x^{\prime }$ of Equation (4) such that

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

(9)

Taking the difference and sum of Equations (7) and (9), one obtains that

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

(10)

and

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

(11)

From the first equation in (10), we have

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

(12)

Inserting Equation (12) into Equations (10) and (11) yields the Bäcklund transformation

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

(13)

When taking $v=0$ in Equation (2), it becomes

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

(14)

A solution to (14) can be taken as

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

(15)

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

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

(16)

where

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

(17)

A specific choice of ${\mathrm{\Lambda }}_l$ will result in special solutions. Then, we consider two different cases. When ${\mathrm{\Lambda }}_l={\delta }_{l,l_1}+{\delta }_{l,l_2}$, from (16), one infers that

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

(18)

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

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

(19)

as shown in Figure 1a.

When ${\mathrm{\Lambda }}_l={\delta }_{l,l_1}-{\delta }_{l,l_2}$, from (16), we have

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

(20)

which is a singular solution; see Figure 1b.

3. A Type of New Soliton Solutions

Spectral problem (14) possesses not only soliton solutions such as (18) and (20) but also the following solution:

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

(21)

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

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

(22)

where

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

(23)

Therefore, a new solution to Equation (2) (see Figure 2) is given by

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

(24)

When ${\mathrm{\Lambda }}_l={\delta }_{l,l_1}+{\delta }_{l,l_2}$, Equation (21) becomes

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

(25)

Similarly to the above calculation, we obtain another new solution to Equation (2):

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

(26)

Based on soliton solution (16), we construct a new solution form (21), which in turn yields new soliton solutions (24) and (26) for the 3dKP equation, which can better reveal the nonlinear dynamical behavior of the 3dKP equation.

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 obtain the superposition formula, let $v_1$ be a solution given by a Bäcklund transformation from a known solution $v_0$ with some spectral function ${\mathrm{\Lambda }}_{1,l}$, $v_2$ be a second solution generated from $v_0$ with spectral function ${\mathrm{\Lambda }}_{2,l}$, and $v_3$ be a third solution obtained from $v_1$ with spectral functions ${\mathrm{\Lambda }}_{2,l}$. According to the definition and Bäcklund transformation (13), one obtains that

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

(27)

Form (27), we can deduce that

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

(28)

which is just the superposition formula. Starting from $v_0=0$, we have $v_1$ and $v_2$ as given in Equations (18) and (20) with spectral functions ${\mathrm{\Lambda }}_{1,l}$ and ${\mathrm{\Lambda }}_{2,l}$, respectively, which can be referred to as single-spectrum solutions. Substituting them into Formula (28), we can obtain a solution $v_3$ that contains two spectral function ${\mathrm{\Lambda }}_{1,l}$ and ${\mathrm{\Lambda }}_{2,l}$. In terms of the calculation, we can obtain many new solutions to Equation (4).

5. Conclusions

In this paper, the Bäcklund transformation (13) of the 3dKP was constructed by means of the Lax pair (3), 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 the Bäcklund transformation method may be a future research topic. Various types of exact solutions to KP equation (1) in three spatial dimensions were analyzed. The methods and results presented in this paper may provide good inspiration for dealing with similar high-dimensional nonlinear equations [20,21,22,23,24].