Multiple soliton solutions and similarity reduction of a (2+1)-dimensional variable-coefficient Korteweg–de Vries system

Nonlinear partial differential equations play a crucial role in modeling wave phenomena that arise in various fields, such as fluid mechanics, nonlinear optics, plasma physics, condensed matter physics, etc. Methods for constructing their analytical solutions, that can explicitly describe the dynamical behavior, have attracted much attention, among which there are the Hirota’s bilinear method [18, 21], Darboux transformation [16, 47], the inverse scattering transformation [2, 41], the multiple exp-function method [26], dressing method [29, 22], symmetry method [20, 33, 19, 6, 37], and so on. In the current paper, we will mainly be focused on soliton solutions by Hirota’s bilinear method and similarity solution by symmetry reduction.

Hirota’s bilinear method has been investigated by many scholars (see, e.g., [13, 38]). For integrable nonlinear evolution equations, Hirota’s bilinear method can be used to construct $N$-soliton solutions applying the superposition of solutions, and has been extended to obtain breather, lump and their interaction solutions [24], higher-order localized wave [14], rational and semi-rational solution [17]. It was also employed for searching the localized waves of nonlocal evolution equations [48, 49] and variable-coefficient evolution equations [23, 36, 45].

On the other hand, continuous symmetries have also been widely applied to the analysis of differential equations (see, e.g., [7, 34, 35]). They can lead to exact solutions or reductions of differential equations [33, 6], and are closely relevant to their integrability (see, e.g., [30]). In recent years, symmetry analysis of variable-coefficient evolution equations has attracted much attention. For instance, in [32, 31], Mohamed and co-authors investigated lump soliton, solitary waves and exponential solutions of the (3+1) dimensional variable-coefficients Kudryashov–Sinelshchikov equation and the (2+1)-dimensional variable-coefficient Bogoyavlensky–Konopelchenko equation by similarity reduction using their symmetries. In [44], a variable-coefficient Davey–Stewartson system was studied, where the optimal system of symmetries was obtained with adjoint representation. Variable-coefficient Davey–Stewartson system that admits the Kac-Moody-Virasoro symmetry was proposed in [15]. In [46], nonlocal symmetries of the coupled variable-coefficient Newell–Whitehead system were used to calculate its group-invariant solutions. However, few studies have been conduced on the dynamics of higher-order localized waves. In the current paper, we focus on nonlinear wave structures of the following (2+1)-dimensional variable-coefficient Korteweg–de Vries (KdV) system

$$\left\{\begin{array}[]{l}\rho(t)u_{t}+3\mu(t)(uv)_{x}+\sigma(t)u_{xxx}=0,\vspace{0.2cm}\\ u_{x}=v_{y},\end{array}\right.$$

(1.1)

and particularly its integrable variant with both $\mu(t)$ and $\sigma(t)$ constant functions, using both Hirota’s bilinear method and symmetry analysis. Here, $u$, $v$ are the dependent variables and $x$, $y$ and $t$ are the independent variables. The notations $u_{t}=\partial u/\partial t$, $u_{x}=\partial u/\partial x$ and so forth are adopted in the current paper. The functions $\rho(t)$, $\mu(t)$ and $\sigma(t)$ are known but arbitrary functions of $t$ which are assumed to be smooth enough; when $\rho(t)=\mu(t)=\sigma(t)=1$, the system (1.1) reduces to the constant-coefficient (2+1)-dimensional KdV system [42, 25], derived by Boiti et al. using the weak Lax pair [11]. Furthermore, the system (1.1) reduces to the (1+1)-dimensional KdV equation by setting $v=u$ and $x=y$.

In Section 2, integrability condition of (1.1) is analyzed by Painlevé analysis and in what follows we will be focused on its integrable version with constant coefficients $\mu$ and $\sigma$. In Section 3, $N$-soliton solutions and hybrid interaction of line solitons, and breather and lump solitons are obtained through Hirota’s bilinear method, while in Section 4, we invoke the symmetry method to derive Lie point symmetries and obtain the corresponding similarity solutions. In particular, a PDE (see Eq. (4.16) or system (4.14) in the potential form below) passing the Painlevé test is derived by symmetry reduction, that reads

$\displaystyle U_{r}=\left(\frac{a(y)-\sigma U_{rr}}{3\mu U}\right)_{y},$

(1.2)

where $r,y$ are the independent variables and $U$ is the dependent variable, and $a(y)$ is an arbitrary function. Further symmetry reduction shows that it can be reduced to non-autonomous third-order Painlevé equations, which are in the form of Chazy’s classification of third-order ODEs by Painlevé analysis but not included in Chazy’s 13 classes. The final Section 5 is dedicated to conclusion.

In this section, we study integrability condition of the (2+1)-dimensional variable-coefficient KdV system (1.1) through Painlevé analysis (see, e.g., [40, 43]). Let $u=m_{y}$ and $v=m_{x}$, and the system (1.1) becomes

$$\rho(t)m_{yt}+3\mu(t)(m_{x}m_{y})_{x}+\sigma(t)m_{xxxy}=0,$$

(2.1)

that is assumed to admit a solution as a Laurent expansion about a singular manifold $\phi=\phi(x,y,t)$ as follows

$$m(x,y,t)=\phi^{-n}(x,y,t)\sum_{j=0}^{\infty}m_{j}(x,y,t){\phi^{j}(x,y,t)},\quad n>0.$$

(2.2)

Here, $n$ is determined by a leading-term analysis and balancing the dominant terms $(m_{x}m_{y})_{x}$ and $m_{xxxy}$. Straightforward calculation gives $n=1$ and

$$m_{0}=\frac{2\sigma(t)}{\mu(t)}\phi_{x}.$$

(2.3)

Substituting $m=m_{0}\phi^{-1}+q\phi^{r-1}$ back to Eq. (2.2) and balancing the most singular term again, we obtain

$$q\Big{(}3\mu(t)(4m_{0}(r-1)\phi_{x}^{2}\phi_{y}-2m_{0}(r-1)(r-2)\phi_{x}^{2}\phi_{y})+\sigma(t)(r-1)(r-2)(r-3)(r-4)\phi_{x}^{3}\phi_{y}\Big{)}=0.$$

(2.4)

Combining Eqs. (2.3) and (2.4), $q$ is arbitrary when $r=-1,1,4$ and $6$, which are the resonant points. Substitution of (2.2) into Eq. (2.1) then amounts to the recursion relations for the $m_{j}$, which take the form of coupled partial differential equations. Finally, we observe that explicit expressions for $m_{2}$, $m_{3}$ and $m_{5}$ and compatibility condition to ensure integrability requires $\mu(t)=\mu$, $\sigma(t)=\sigma$ to be non-zero constants, but $\rho(t)$ remains to be an arbitrary function of $t$. Consequently, $m_{1}$, $m_{4}$ and $m_{6}$ are arbitrary functions.

To summarize up, we conclude that the following special (2+1)-dimensional time-dependent variable-coefficient KdV system

$$\left\{\begin{array}[]{l}\rho(t)u_{t}+3\mu(uv)_{x}+\sigma u_{xxx}=0,\vspace{0.2cm}\\ u_{x}=v_{y},\end{array}\right.$$

(2.5)

is integrable as passes the Painlevé test, where $\mu$ and $\sigma$ are constants. In the rest of the paper, we will be focused on studies of analytic solutions of the integrable system (2.5).

The integrable (2+1)-dimensional time-dependent variable-coefficient KdV system (2.5) can be written in bilinear form

$$\Big{(}\rho(t)D_{y}D_{t}+\sigma D_{x}^{3}D_{y}+3\alpha\mu D_{x}^{2}+3\beta\mu D_{x}D_{y}\Big{)}f\cdot f=0$$

(3.1)

by the transformation

$$\left\{\begin{array}[]{l}u=\alpha+\frac{2\sigma}{\mu}\big{(}{\rm log}(f)\big{)}_{xy},\vspace{0.2cm}\\ v=\beta+\frac{2\sigma}{\mu}\big{(}{\rm log}(f)\big{)}_{xx},\\ \end{array}\right.$$

(3.2)

where both $\alpha$ and $\beta$ are real-valued parameters, and Hirota’s bilinear differential operators are defined by

$\displaystyle{}D_{x}^{m}D_{y}^{n}D_{t}^{s}a\cdot b=(\partial_{x}-\partial_{x^{\prime}})^{m}(\partial_{y}-\partial_{y^{\prime}})^{n}(\partial_{t}-\partial_{t^{\prime}})^{s}a(x,y,t)b(x^{\prime},y^{\prime},t^{\prime})|_{x=x^{\prime},y=y^{\prime},t=t^{\prime}}.$

(3.3)

The function $f$ can have the general form as

$\displaystyle{}f_{N}=\sum_{\varsigma=0,1}\exp\left(\sum_{j=1}^{N}\varsigma_{j}\theta_{j}+\sum_{1\leq s<j}^{N}\varsigma_{s}\varsigma_{j}A_{sj}\right),\quad N=1,2,\ldots,$

(3.4)

where

$\displaystyle{}\theta_{j}=k_{{j}}x+p_{{j}}y-\omega_{j}(t)+\theta_{0j}\text{ with }\omega_{j}(t)=\int\frac{\sigma k_{j}^{3}p_{j}+3\beta\mu k_{j}p_{j}+3\alpha\mu k_{j}^{2}}{p_{j}\rho(t)}dt$

(3.5)

for $j=1,2,\ldots,N$, and

$\displaystyle{}\exp(A_{sj})=\frac{\sigma k_{s}k_{j}(k_{s}-k_{j})(p_{s}-p_{j})p_{s}p_{j}-\alpha\mu(k_{j}p_{s}-k_{s}p_{j})^{2}}{\sigma k_{s}k_{j}(k_{s}+k_{j})(p_{s}+p_{j})p_{s}p_{j}-\alpha\mu(k_{j}p_{s}-k_{s}p_{j})^{2}},~{}~{}1\leq s<j\leq N.$

(3.6)

Here, $k_{j},p_{j},\theta_{0j}$ are arbitrary constants.

Substituting the function $f$ in (3.4) and (3.5)-(3.6) into the transformation in (3.2), the $N$-solitons of the (2+1)-dimensional time-dependent variable-coefficient KdV system (2.5) can be constructed explicitly.

Two-soliton solution. When $N=2$, Eq. (3.4) reads

$\displaystyle{}f_{2}=1+\exp(\theta_{1})+\exp(\theta_{2})+\exp(\theta_{1}+\theta_{2}+A_{12}),$

(3.7)

and the two-solitons of Eq. (2.5) can be obtained via (3.2). This covers the results of [12] by specifying $\sigma=1,\ \ \mu=1$ and $\rho(t)\equiv 1$. In the following, we always take $\sigma=1,\ \ \mu=1$ and consider various functions $\rho(t)$. Fig. 1 shows the two-soliton solution with $\rho(t)\equiv 1$. The interacting line solitons form H-type and X-type, which were observed in ocean waves [1]. Both H-type and X-type interaction with long stem, the wave form $u$ have similar structure, while for $v$, the stem of H-type has a lower amplitude and X-type has the opposite result.

Three-soliton solution. When $N=3$, Eq. (3.4) becomes

$\displaystyle{}\begin{split}f_{3}&=1+\exp(\theta_{1})+\exp(\theta_{2})+\exp(\theta_{3})+\exp(\theta_{1}+\theta_{2}+A_{12})\\ &\quad+\exp(\theta_{1}+\theta_{3}+A_{13})+\exp(\theta_{2}+\theta_{3}+A_{23})+\exp(\theta_{1}+\theta_{2}+\theta_{3}+A_{123}),\end{split}$

(3.8)

where $A_{123}=A_{12}A_{13}A_{23}$. Substituting $f_{3}$, i.e., Eq. (3.8), to (3.2), three-soliton solution can be obtained. Fig. 2 shows novel wave structure with respect to various variable coefficients $\rho(t)$. Top and bottom of the figure illustrate the 3d plots of $u$ and $v$, respectively. Figs. 2 (a1)-(d2) are plotted with the same parameters $\alpha=-0.5,\beta=-0.5$, $k_{1}=1,k_{2}=2,k_{3}=2,p_{1}=2,p_{2}=2$, but different $p_{3}$: (a1) (a2) $p_{3}=-2$, (b1) (b2) $p_{3}=-2$, (c1) (c2) $p_{3}=4$, (d1) (d2) $p_{3}=3$. It is observed that shapes of waves are influenced by the variable coefficient $\rho(t)$.

• •

  For $\rho(t)=1/t$, the three-soliton solution shows the interaction among three parabolic solitons (see Figs. 2 (a1) and (a2)).

• •

  For $\rho(t)=1/{t^{2}}$, the shape of the wave changes from parabolic to cubic, as shown by Figs. 2 (b1) and (b2).

• •

  For trigonometric functions, periodic waves are obtained. See Figs. 2 (c1), (c2) for $\rho(t)=1/\sin t$ and Figs. 2 (d1), (d2) for $\rho(t)=1/\left(\sin 2t+\tanh t\right)$.

By specifying the conjugate parameters, two linear solitons can be reduced to one breather. By applying the long wave limit method, two linear solitons can be reduced to one lump solution [28]. The following theorem can then be obtained.

Let $M=1,L=1,Q=1$ in (3.1), and so $N=5$. Namely, five-soliton solution can be reduced to the interaction among one lump, one breather and one line soliton. Figs. 3 (a1), (b1) describe the dynamical behavior in the $(y,t)$- plane when $x=2$, while Figs. 3 (c1), (d1) describe the dynamical behavior in the $(x,t)$- plane when $y=2$. It is noticed that the lump, breather and line solitons are localized in the parabolic curves and interact with each other.

In the following, we will show several other special $N$-soliton solutions as corollaries of Theorem 3.1.

Now, the corresponding representation can be obtained from (3.2), if we set

$${}f_{2L}=\sum_{\varsigma=0,1}\exp\left(\sum_{j=1}^{2L}\varsigma_{j}\theta_{j}+\sum_{1\leq s<j}^{2L}\varsigma_{s}\varsigma_{j}A_{sj}\right),$$

(3.10)

with $\theta_{j}=\kappa_{j}+i\lambda_{j},$ and $\exp(A_{sj})$ defined by (3.5) and (3.6) respectively, and consequently

$$\begin{split}&\zeta_{j}=\delta_{j}x+\kappa_{j}y-\int\frac{\Lambda_{j}}{\kappa_{j}^{2}+\lambda_{j}^{2}}dt+\zeta_{0j},\\ &\xi_{j}=\gamma_{j}x+\sigma_{j}y-\int\frac{\Upsilon_{j}}{\kappa_{j}^{2}+\lambda_{j}^{2}}dt+\xi_{0j},\end{split}$$

(3.11)

where $j=1,2,\ldots,L$, and

$$\begin{split}&\Lambda_{j}=\sigma(\kappa_{j}^{2}+\lambda_{j}^{2})\delta_{j}^{3}+3\alpha\mu\delta_{j}^{2}\kappa_{j}+3\Big{(}(\beta\mu-\lambda_{j}^{2}\sigma)\kappa_{j}^{2}+(2\alpha\lambda_{j}\gamma_{j}+\beta\lambda_{j}^{2})\mu-\sigma\lambda_{j}^{2}\gamma_{j}^{2}\Big{)}\delta_{j}-3\alpha\mu\lambda_{j}^{2}\kappa_{j},\\ &\Upsilon_{j}=-\sigma(\kappa_{j}^{2}+\lambda_{j}^{2})\gamma_{j}^{3}+3\alpha\mu\gamma_{j}^{2}\lambda_{j}+3\Big{(}(\delta_{j}^{2}\sigma+\beta\mu)\lambda_{j}^{2}+(2\alpha\kappa_{j}\delta_{j}+\kappa_{j}^{2}\beta)\mu+\sigma\kappa_{j}^{2}\delta_{j}^{2}\Big{)}\gamma_{j}-3\alpha\mu\delta_{j}^{2}\lambda_{j}.\end{split}$$

When $L=2$, the four-soliton solution reduces to the interaction between two breathers for $\rho(t)=1/t^{2}$. Figs. 4 (a1) and (b1) describe the dynamical behavior in the $(y,t)$- plane when $x=0$, while Figs. 4 (c1) and (d1) describe the dynamical behavior in the $(x,t)$- plane when $y=0$.We notice that the two breathers have different amplitudes and periods, but both are with an S-type structure.

The corresponding representation can be obtained from (3.2) by using

$\displaystyle{}\begin{split}f_{2M}&=\prod_{j=1}^{2M}\Theta_{j}+\frac{1}{2}\sum_{s,j}^{2M}a_{sj}\prod_{l\neq s,j}^{2M}\Theta_{l}+\frac{1}{2!2^{2}}\sum_{s,j,k,m}^{2M}a_{sj}a_{km}\prod_{l\neq s,j,k,m}\Theta_{l}+\cdots\\ &\quad+\frac{1}{M!2^{M}}\sum_{s,j,k,m}^{2M}a_{sj}\overbrace{a_{rl}\cdots a_{wn}}^{M}\prod_{p\neq s,j,r,l,\ldots,w,n}^{2M}\Theta_{p}+\cdots,\end{split}$

(3.12)

where

$\displaystyle\begin{split}&\Theta_{s}=x+p_{s}y-3\mu\int\frac{\beta p_{s}+\alpha}{\rho(t)p_{s}}dt,\quad s=1,2,\ldots,2M,\\ &a_{sj}=\frac{2\sigma p_{s}p_{j}(p_{s}+p_{j})}{\alpha\mu(p_{s}-p_{j})^{2}},\quad 1\leq s<j\leq 2M,\end{split}$

(3.13)

with $p_{s}$ ($s=1,2,\ldots,2M$) arbitrary complex constants. Let $M=2$ in (3.12). Fig. 5 shows that the four-soliton solution reduces to the interaction between two lump solutions. The effect of the variable coefficient on the interactions is provided on the $(y,t)$-plane for $x=0$, i.e., (a1), (a2), (b1), (b2), and $(x,t)$-plane for $y=0$, i.e., (c1), (c2), (d1), (d2). Obviously the variable coefficient $\rho(t)$, chosen as $1/t$ in Fig. 5, is closely related to wave shapes.

In this section, we conduct symmetry analysis and in particular similarity reduction of the (2+1)-dimensional integrable variable-coefficient KdV system (2.5).