1 Introduction
The investigation of exact solutions to nonlinear partial differential equations is one of the most important problems. Many kinds of soliton solutions are studied by a variety of methods including the inverse scattering transformation,[1] the Darboux transformation,[2-3] the Hirota bilinear method,[4] and symmetry reductions,[5] etc.[6-10] Recently, lump solutions which are rational, analytical and localized in all directions in the space,[11-20] have attracted much attention. As another kind of exact solutions, it exsits potential applications in physics, partically in atmospheric and oceanic sciences.[21]
The Hirota bilinear method in soliton theory provides a powerful approach to finding exact solutions.[4] A kind of lump solutions can be also obtained by means of the Hirota bilinear formuation. Recently, the generalized bilinear operators are proposed by exploring the linear superposition principle.[22] Many new nonlinear systems are constructed by using the generalized Hirota bilinear operators.[23-26] The lump solutions and integrable propertites for those new nonlinear systems are interesting topic in nonlinear science.
The paper is organized as follows. In Sec.~2, a new nonlinear differential equation is constructed by means of the bilinear formulation. The new nonlinear equation includes a Calogero-Bogoyavlenskii-Schiff equation and a Bogoyavlensky-Konopelchenko (gCBS-BK) equation. A class of gCBS-BK-like equations can be obatined by using the generalized bilinear method. In Sec.~3, a lump solution to the newly presented gCBS-BK systems is obtained bsaed on the $Maple$ symbolic computations. Two figures are given theoretically and graphically. The last section is devoted to summary and discussions.
2 A Generalized gCBS-BK Equation
We consider a (2+1)-dimensioanl nonlinear partial differential equation
$\begin{matrix} && u_{t} + u_{xxy} + 3u_xu_{y} + \delta_1 u_{y} + \delta_2 w_{yy} + \delta_3u_x \nonumber\\ && \;\; + \,\delta_4 (3u_x^2+ u_{xxx}) + \delta_5(3w_{yy}^2 + w_{yyyy}) \nonumber\\ &&\;\; + \,\delta_6 (3u_{y}w_{yy} + u_{yyy}) = 0\,, \quad u_x=w\,,\label{1} \end{matrix}$
where $\delta_i, i=1,2,\ldots,6$ are arbitrary constants. While the constants satisfy $\delta_3= \delta_4 = \delta_5=\delta_6=0$ and $\delta_5=\delta_6=0$, (1) becomes a generalized Calogero-Bogoyavlenskii-Schiff (CBS) equation[18] and a generalized[18,27] Bogoyavlensky-Konopelchenko (BK) equation,[19] respectively. The CBS equation was constructed by the modified Lax formalism and the self-dual Yang-Mills equation respectively.[28-29] The BK equation is described as the interaction of a Riemann wave propagating along $y$-axis and a long wave propagating along $x$-axis.[30] These two equations have been widely studied in different ways.[31-32] The (2+1)-dimensional nonlinear differential equation (1) is thus called gCBS-BK equation. The Hirota bilinear form of gCBS-BK equation (1) has
$\begin{matrix} && \;\;\;D_t D_x +D_x^3D_y + \delta_1 D_x D_y + \delta_2 D_y^2 +\delta_3 D_x^2 + \delta_4 D_x^4 + \delta_5 D_y^4 + \delta_6 D_xD_y^3 \nonumber \\ && = 2(f_{xt}f-f_tf_x + f_{xxxy}f - f_{xxx}f_y - 3f_{xxy}f_x + 3f_{xx}f_{xy} + \delta_1 (f_{xy}f-f_xf_y) \nonumber \\ &&\;\;\; + \,\delta_2 (f_{yy}f-f_y^2) + \delta_3 (f_{xx}f-f_x^2) + \delta_4 (f_{xxxx}f - 4f_xf_{xxx}-3f_{xx}^2) \nonumber \\ && \;\;\;+ \,\delta_5(ff_{yyyy} -4f_yf_{yyy}+3f_{yy}^2) + \delta_6(ff_{xyyy}-f_xf_{yyy}-3f_yf_{xyy}+3f_{xy}f_{yy})) =0, \label{sysd} \end{matrix}$
by the relationship between $u, w$, and $f$
$\begin{equation}w = 2(\ln f)_{xx}= \frac{2(f_{xx}f-f_x^2)}{f^2}, \hspace{0.8cm} u=2(\ln f)_x = \frac{2f_x}{f}. \label{syst}\end{equation}$
Based on the generalized bilinear thoery,[22] the generalized bilinear operators read
$\begin{matrix} && (D_{p,x}^mD_{p,t}^n)f(x,t)\cdot f(x', t') = (\partial_x + \alpha_p \partial_{x'})^m (\partial_t + \alpha_p \partial_{t'})^n f(x,t)f(x', t')|_{x'=x,t'=t} \nonumber \\ && = \sum_{i=0}^m \sum_{j=0}^n \left ( \begin{matrix} m \\ i \end{matrix} \right ) \left ( \begin{matrix} n \\ j \end{matrix} \right ) \alpha_p^i \alpha_p^j \frac{\partial^{m-i}}{\partial x^{m-i}} \frac{\partial^i}{\partial x^{'(i)}} \frac{\partial^{n-j}}{\partial t^{n-j}} \frac{\partial^j}{\partial t^{'(j)}} f(x,t)f(x',t')|_{x'=x,t'=t} \nonumber \\ && = \sum_{i=0}^m \sum_{j=0}^n \left ( \begin{matrix} m \\ i \end{matrix} \right ) \left ( \begin{matrix} n \\ j \end{matrix} \right ) \alpha_p^i \alpha_p^j \frac{\partial^{m+n-i-j} f(x,t)}{\partial x^{m-i} \partial t^{n-j}} \frac{\partial^{i+j}f(x,t)}{\partial x^{i} \partial t^j}, \end{matrix}$
where $m, n \geq 0$ and $\alpha_p^s=(-1)^{r_p(s)}$ if $s=r_p(s)$ mod $p$. Here $\alpha_p$ is a symbol. For a prime number $p>2$, we can not write the relationship
$\begin{matrix} \alpha_p^i\alpha_p^j=\alpha_p^{i+j}, \hspace{0.5cm} i,j \geq 0. \end{matrix}$
Taking the prime number $p=3$, we have
$\begin{matrix} \alpha_3=-1, \hspace{0.5cm} \alpha_3^2=1, \hspace{0.5cm} \alpha_3^3=1, \hspace{0.5cm} \alpha_3^4=-1, \hspace{0.5cm} \alpha_3^5=1, \hspace{0.5cm} \alpha_3^6=1, \hspace{0.5cm} \ldots, \end{matrix}$
and then, we have the concrete operators
$\begin{matrix} && D_{3,t} D_{3,x}f\cdot f= 2f_{xt}f-2f_xf_t\,, \quad D_{3,x}^3D_{3,y}= 6f_{xx}f_{xy}\,, \quad D_{3,x} D_{3,y}=2f_{xy}f-2f_xf_y\,, \nonumber\\ && D_{3,y}^2= 2f_{yy}f-2f_y^2\,,\quad D_{3,x}^2= 2f_{xx}f-2f_x^2\,, \quad D_{3,x}^4= 6f_{xx}^2\,,\quad D_{3,y}^4= 6f_{yy}^2\,,\quad D_{3,x}D_{3,y}^3= 6f_{yy}f_{xy}\,. \end{matrix}$
By the above analysis, the corresponding bilinear form of the gCBS-BK equation (1) in $p=3$ reads
$\begin{matrix} && \;\;\;D_{3,t} D_{3,x} +D_{3,x}^3D_{3,y} + \delta_1 D_{3,x} D_{3,y} + \delta_2 D_{3,y}^2 + \delta_3 D_{3,x}^2 + \delta_4 D_{3,x}^4 + \delta_5 D_{3,y}^4 + \delta_6 D_{3,y}D_{3,y}^3 \nonumber \\ && = 2(f_{xt}f-f_tf_x + 3f_{xx}f_{xy} + \delta_1 (f_{xy}f-f_xf_y) + \delta_2 (f_{yy}f-f_y^2) + \delta_3 (f_{xx}f-f_x^2) \nonumber\\ && \;\;\; + \, 3\delta_4 f_{xx}^2 + 3\delta_5 f_{yy}^2 +3\delta_6f_{yy}f_{xy}) =0|\,. \label{sysdd} \end{matrix}$
Bell polynomial theories suggest a dependent variable transfomation
$\begin{matrix} u= 2(\ln f)_x, \label{tras}\end{matrix}$
to transfrom bilinear equations to nonlinear equations. By selecting the variable transformation (9), a gCBS-BK-like equation is obtained from the generalized bilinear form (8)
$\begin{matrix} && u_{t} + \frac{3}{4}u^2 u_y + \frac{3}{2} u_xu_y + \frac{3}{4} uu_x w_y + \frac{3}{8} u^3w_y \nonumber\\ && \;\;+\, \delta_1 u_y + \delta_2 w_{yy} + \delta_3u_x + \frac{3}{8}\delta_4 (u^2+2u_x)^2 \nonumber\\ && \;\;+ \,\frac{3}{2}\delta_5 (w_y^2+w_{yy})^2 +\frac{3}{8}\delta_6(w_y^2+w_{yy})(uw_y + 2u_y) = 0\,,\nonumber \\ && u_x=w\,. \label{gcsf} \end{matrix}$
By selecting the prime number $p=3$, we get a new gCBS-BK-like equtaion (10). We can aslo select $p=5,7,9,\ldots$ to get new nonlinear partial differential equations. This provides a useful method to get new nonlinear systems that possess bilinear forms. In this paper, we shall focus on the gCBS-BK equation (1) and the gCBS-BK-like equation (10) for the prime number $p=3$.
3 A Search for Lump Solution
Based on the bilinear form, a quadratic function solution to the (2+1)-dimensional bilinear gCBS-BK equation (2) and bilinear gCBS-BK-like equation (8), is defined by
$\begin{matrix} && f= \xi _1^2 + \xi_2^2 + a_9\,, \nonumber\\ && \xi_1= a_1 x+a_2 y +a_3t +a_4\,, \nonumber \\ && \xi_2= a_5 x+a_6y+a_7 t + a_8\,, \label{solgf} \end{matrix}$
where $a_i$, $1 \leq i \leq 9$ are constant parameters to be determined. Substituting the expression (11) into Eqs. (2) and (8) and vanishing the coefficients of different powers of $x, y$, and $t$, we can get the same relationship among parameters for Eqs. (2) and (8). The following set of solutions for the parameters $a_3$, $a_7$, and $a_9$
Fig.1 (Color online) Profiles of the lump solution (13). (a) 3D lump plot with the time $t=0$, (b) the corresponding density plot, (c) the curve by selecting different parameters $y$ and $t$, (d) the curve by selecting different parameters $x$ and $t$. |
Fig.2 (Color online) Profiles of the lump solution (14). (a) 3D lump plot with the time $t=0$, (b) the corresponding density plot, (c) the curve by selecting different parameters $y$ and $t$, (d) the curve by selecting different parameters $x$ and $t$. |
$\begin{matrix} && a_3 = -\delta_1a_2 -\delta_3 a_1 - \frac{\delta_2(a_1a_2^2-a_1a_6^2+2a_2a_5a_6)}{a_1^2+a_5^2}, \nonumber \\ && a_7= -\delta_1 \Big(\frac{a_1a_2}{a_5} + a_6\Big) - \delta_2\frac{(a_1a_2 + a_5a_6)^2-(a_1a_6-a_2a_5)^2}{a_5(a_1^2+a_5^2)} - \delta_3\frac{a_1^2+a_5^2}{a_5} - \frac{a_1a_3}{a_5}, \nonumber \\ && a_9= - \frac{3(a_1^2+a_5^2)^2}{\delta_2(a_1a_6-a_2a_5)^2} \Bigl(a_1a_2+a_5a_6 + \delta_4(a_1^2+a_5^2) + \delta_5\frac{(a_2^2+a_6^2)^2}{a_1^2+a_5^2} + \delta_6\frac{(a_2^2+a_6^2)(a_1a_2+a_5a_6)}{a_1^2+a_5^2}\Bigr), \label{socns} \end{matrix}$
which need to satisfy the following conditions
(i) $ a_5 \neq 0$, to guarantee the well-posedness for $f$;
(ii) ${\delta_2}\Big(a_1a_2+a_5a_6 + \delta_4(a_1^2+a_5^2) + \delta_5\dfrac{(a_2^2+a_6^2)^2}{a_1^2+a_5^2} + \delta_6\dfrac{(a_2^2+a_6^2)(a_1a_2+a_5a_6)}{a_1^2+a_5^2}\Big) < 0$, to have the positivity of $f$;
(iii) $a_1a_6 - a_2a_5 \neq 0$, to ensure the localization of $u, w$ in all directions in the space.
The parameters take $a_1=1, a_2=-2, a_4=-2, a_5=-2, a_6=2, a_8=1, \delta_1=1, \delta_2=1, \delta_3=1,\delta_4=1, \delta_5=1, \delta_6=2$. By substituting Eq. (11) into Eq. (9) and combining the relationship (12), we get the lump solution
$\begin{matrix} u=-\frac{16(27t-25x+30y+20)}{100x^2+160y^2-240xy+240y-160x + 240t+304yt-216tx+148t^2 + 2875}\,. \label{sofi} \end{matrix}$
The 3D plot, density plot, and curve plot for this lump solution are depicted in Fig.1. The parameters take $a_1=1$, $a_2=1$, $a_4=1$, $a_5=-2$, $a_6=3$, $a_8=1$, $\delta_1=1$, $\delta_2=1$, $\delta_3=1$,$\delta_4=1$, $\delta_5=-2$, $\delta_6=2$. The lump solution has the following form
$\begin{matrix}u=\frac{4(5x-5y-1)}{5x^2+10y^2-10xy+8y-2x+ 6t+10yt+5t^2 + 182}\,. \label{sose}\end{matrix}$
The 3D plot, density plot and curve plot for the lump solution are shown in Fig.2.
4 Summary and Discussions
In summary, the gCBS-BK equation was derived in terms of Hirota bilinear forms. By selecting the prime number $p=3$, a gCBS-BK-like equation was formulated by the generalized Hirota operators. The lump solution of the gCBS-BK equation and the gCBS-BK-like equation was generated by their Hirota bilinear forms. The phenomena of lump solutions were presented by figures. The results provide a new example of (2+1)-dimensional nonlinear partial differential equations, which possess lump solutions. Other new nonlinear equations can be also obtained by seleting the prime numbers $p=5, 7, \ldots$ It is demonstrated that the generalized Hirota operators are very useful in constructing new nonlinear differential equations, which possess nice math properties. In the meanwhile, lump-kink interaction solutions,[34-35] lump-soliton interaction solutions,[36] lump type solutions for the (3+1)-dimensional nonlinear differential equations[36-38] and solitons-cnoidal wave interaction solutions[39-41] are important and will be explored in the future.