Similarity reduction and multiple novel travelling and solitary wave solutions for the two-dimensional Bogoyavlensky–Konopelchenko equation with variable coefficients

Abstract

In this study, the two-dimensional Bogoyavlensky–Konopelchenko equation with variable coefficients (vcBK) which describes the interaction of Riemann waves has been solved two times by using the balance technique. First, the balance procedure was used to reduce the vcBK equation into a fourth-order nonlinear ordinary differential equation and by its implementation again composed with the auxiliary method, many novel periodic, kink and solitary wave solutions were obtained for the vcBK equation. Second, the balance method was used to construct the Bäcklund transformation for the vcBK equation, by which n-soliton solutions were obtained. Finally, some figures were given to show the soliton interaction for different values of the variable coefficients.

Keywords:

• 2D Bogoyavlensky–Konopelchenko equation with variable coefficients
• direct similarity reduction method
• Bäcklund transformation
• N-soliton solutions
• travelling wave solutions

1. Introduction

Problems of physical interest are often translated in terms of differential equations that may turn out to be linear or nonlinear, ordinary or partial. Analytical solutions of these resulting equations are of much interest both from mathematical and application points of view. On account of their applications in such conventional areas as physics and engineering and most recent applications in the disciplines of biology, chemistry, ecology and economics, differential equations have retained their central role. This is also evident through our recent fascination for the theory of solitons. However, much of the current interest in differential equations is due to the advent of high speed computing facilities and the consequent ability to shift emphasis from the classical study of linear systems to the fascinating problems encountered in the study of nonlinear systems. Finding soliton-type solutions for nonlinear partial differential equations (NPDE) is an important mission for mathematicians many new techniques were investigated for that like Hirota bilinear method, Bäcklund transformations, symmetry groups, etc. [1–26].

A new challenge appears by trying to obtain solitary wave solutions for generalized and complicated forms of NPDEs specially, the nonlinear evolution equations with variable coefficients, because it reflects the real physical situation than the constant version and have various applications in many fields of science like optics, plasma physics, etc.

One of the new variable coefficients of nonlinear evolution equations is the variable coefficient Bogoyavlensky–Konopelchenko (vcBK) equation [27–31] (1) $\begin{array}{rl}& {\phi }_{\mathrm{xt}}+a_1\left(t\right){\phi }_{\mathrm{xxxx}}+a_2\left(t\right){\phi }_{\mathrm{xxxy}}+a_3\left(t\right){\phi }_x{\phi }_{\mathrm{xx}}\\ & +a_4\left(t\right){\phi }_x{\phi }_{\mathrm{xy}}+a_5\left(t\right){\phi }_{\mathrm{xx}}{\phi }_y=0,\end{array}$(1) where $a_i(t),i=1,\dots ,5$ are arbitrary functions of t. The vcBK equation describes the interaction of a Riemann wave propagating along the y-axis and along the x-axis. Wang et al. [27] studied the integrability of Equation (1(1) $\begin{array}{rl}& {\phi }_{\mathrm{xt}}+a_1\left(t\right){\phi }_{\mathrm{xxxx}}+a_2\left(t\right){\phi }_{\mathrm{xxxy}}+a_3\left(t\right){\phi }_x{\phi }_{\mathrm{xx}}\\ & +a_4\left(t\right){\phi }_x{\phi }_{\mathrm{xy}}+a_5\left(t\right){\phi }_{\mathrm{xx}}{\phi }_y=0,\end{array}$(1) ) using Painléve test, they arrived to the following integrable vcBK equation: (2) $\begin{array}{rl}& {\phi }_{\mathrm{xt}}+a\left(t\right){\phi }_{\mathrm{xxxx}}+b\left(t\right){\phi }_{\mathrm{xxxy}}+c\left(t\right){\phi }_x{\phi }_{\mathrm{xx}}\\ & +e\left(t\right)({\phi }_x{\phi }_{\mathrm{xy}}+{\phi }_{\mathrm{xx}}{\phi }_y)=0\end{array}$(2) and after integration for x yielded the vcBK evolution equation as (3) $\begin{array}{r}{\phi }_t+a\left(t\right){\phi }_{\mathrm{xxx}}+b\left(t\right){\phi }_{\mathrm{xxy}}+\frac{c\left(t\right)}{2}{\phi }_x^2+e\left(t\right){\phi }_x{\phi }_y=0.\end{array}$(3)

Equation (3(3) $\begin{array}{r}{\phi }_t+a\left(t\right){\phi }_{\mathrm{xxx}}+b\left(t\right){\phi }_{\mathrm{xxy}}+\frac{c\left(t\right)}{2}{\phi }_x^2+e\left(t\right){\phi }_x{\phi }_y=0.\end{array}$(3) ) solved in [28] where one and two solitons were obtained by the unified method moreover, Lie group method was used in [29–31] and many similarity reductions and solutions were obtained.

In this work, we are going to use the homogeneous balance (HB) method connected with similarity reduction method to obtain a new similarity reduction (compared with other reductions obtained before [29–31]) in a form of fourth-order nonlinear ordinary differential equation and by using the Riccati equation method [22,32–34] multiple novel travelling wave solutions will obtain. Moreover, in Section 3, the HB method will use to construct Bäcklund transformation and n-soliton solutions for the integrable vcBK equation (3(3) $\begin{array}{r}{\phi }_t+a\left(t\right){\phi }_{\mathrm{xxx}}+b\left(t\right){\phi }_{\mathrm{xxy}}+\frac{c\left(t\right)}{2}{\phi }_x^2+e\left(t\right){\phi }_x{\phi }_y=0.\end{array}$(3) ). Finally, in Section 4, the graphical representations for one and two soliton solutions will be given for different values of the variable coefficients, and graphs for kink soliton, parabolic and periodic solitary wave solutions are illustrated.

2. The HB method and similarity reduction

In this section, we are going to apply the HB method connected with the similarity reduction method [35–42] as follows:

Assume that Equation (3(3) $\begin{array}{r}{\phi }_t+a\left(t\right){\phi }_{\mathrm{xxx}}+b\left(t\right){\phi }_{\mathrm{xxy}}+\frac{c\left(t\right)}{2}{\phi }_x^2+e\left(t\right){\phi }_x{\phi }_y=0.\end{array}$(3) ) has a general solution in the form (4) $\begin{array}{r}\phi (x,y,t)={\phi }_0(x,y,t)+\sum _{i=1}^n\varrho \frac{{\mathrm{\partial }}^n{g}^i\left(\eta (x,y,t)\right)}{\mathrm{\partial }{x}^n},\end{array}$(4) where ϱ is a constant to be determined later and $g(\eta (x,y,t))$ is an arbitrary function. By substitution for Equation (4(4) $\begin{array}{r}\phi (x,y,t)={\phi }_0(x,y,t)+\sum _{i=1}^n\varrho \frac{{\mathrm{\partial }}^n{g}^i\left(\eta (x,y,t)\right)}{\mathrm{\partial }{x}^n},\end{array}$(4) ) into Equation (3(3) $\begin{array}{r}{\phi }_t+a\left(t\right){\phi }_{\mathrm{xxx}}+b\left(t\right){\phi }_{\mathrm{xxy}}+\frac{c\left(t\right)}{2}{\phi }_x^2+e\left(t\right){\phi }_x{\phi }_y=0.\end{array}$(3) ), and by balancing the linear terms with the nonlinear terms we get n = 1, so (5) $\begin{array}{r}\phi (x,y,t)={\phi }_0(x,y,t)+\varrho \frac{\mathrm{\partial }g\left(\eta (x,y,t)\right)}{\mathrm{\partial }x}.\end{array}$(5) By using Equation (5(5) $\begin{array}{r}\phi (x,y,t)={\phi }_0(x,y,t)+\varrho \frac{\mathrm{\partial }g\left(\eta (x,y,t)\right)}{\mathrm{\partial }x}.\end{array}$(5) ) into Equation (3(3) $\begin{array}{r}{\phi }_t+a\left(t\right){\phi }_{\mathrm{xxx}}+b\left(t\right){\phi }_{\mathrm{xxy}}+\frac{c\left(t\right)}{2}{\phi }_x^2+e\left(t\right){\phi }_x{\phi }_y=0.\end{array}$(3) ), and collect same terms of g and its derivatives, then we get (6) $\begin{array}{r}\begin{array}{rl}& (a\left(t\right){\eta }_x+b\left(t\right){\eta }_y){\eta }_x^3\varrho g^{\mathrm{\prime \prime \prime \prime }}\\ & +(\frac{1}{2}c\left(t\right){\eta }_x+e\left(t\right){\eta }_y){\eta }_x^3{\varrho }^2{g}^{\mathrm{\prime \prime }2}\\ & +(\frac{1}{2}c\left(t\right){\eta }_{\mathrm{xx}}^2+e\left(t\right){\eta }_{\mathrm{yx}}{\eta }_{\mathrm{xx}}){\varrho }^2{g}^{\mathrm{\prime }2}\\ & ((c\left(t\right){\eta }_{\mathrm{xx}}+e\left(t\right){\eta }_{\mathrm{xy}}){\eta }_x^2+e\left(t\right){\eta }_x{\eta }_y{\eta }_{\mathrm{xx}}){\varrho }^2{g}^{\mathrm{\prime }}{g}^{\mathrm{\prime \prime }}\\ & +((6a\left(t\right){\eta }_{\mathrm{xx}}+3b\left(t\right){\eta }_{\mathrm{xy}}){\eta }_x^2+3b\left(t\right){\eta }_x{\eta }_y{\eta }_{\mathrm{xx}})\varrho g^{\mathrm{\prime \prime \prime }}\\ & +((c\left(t\right){\phi }_{0_x}+e\left(t\right){\phi }_{0_y}){\eta }_x^2+(4a\left(t\right){\eta }_{\mathrm{xxx}}\\ & +3b\left(t\right){\eta }_{\mathrm{xxy}}+e\left(t\right){\phi }_{0_x}{\eta }_y+{\eta }_t){\eta }_x+b\left(t\right){\eta }_y{\eta }_{\mathrm{xxx}}\\ & +3{\eta }_{\mathrm{xx}}(a\left(t\right){\eta }_{\mathrm{xx}}+b\left(t\right){\eta }_{\mathrm{yx}}))\varrho g^{\mathrm{\prime \prime }}+{\phi }_{0_t}+a\left(t\right){\phi }_{0_{\mathrm{xxx}}}\\ & +b\left(t\right){\phi }_{0\mathrm{xxy}}+\frac{1}{2}c\left(t\right){\phi }_{0_x}^2+e\left(t\right){\phi }_{0_x}{\phi }_{0_y}\\ & ({\eta }_{\mathrm{xt}}+a\left(t\right){\eta }_{\mathrm{xxxx}}+b\left(t\right){\eta }_{\mathrm{xxxy}}\\ & +(c\left(t\right){\phi }_{0_x}+e\left(t\right){\phi }_{0_y}){\eta }_{\mathrm{xx}}\\ & +e\left(t\right){\phi }_{0_x}{\eta }_{\mathrm{xy}})\varrho g^{\mathrm{\prime }}=0.\end{array}\end{array}$(6) To make Equation (6(6) $\begin{array}{r}\begin{array}{rl}& (a\left(t\right){\eta }_x+b\left(t\right){\eta }_y){\eta }_x^3\varrho g^{\mathrm{\prime \prime \prime \prime }}\\ & +(\frac{1}{2}c\left(t\right){\eta }_x+e\left(t\right){\eta }_y){\eta }_x^3{\varrho }^2{g}^{\mathrm{\prime \prime }2}\\ & +(\frac{1}{2}c\left(t\right){\eta }_{\mathrm{xx}}^2+e\left(t\right){\eta }_{\mathrm{yx}}{\eta }_{\mathrm{xx}}){\varrho }^2{g}^{\mathrm{\prime }2}\\ & ((c\left(t\right){\eta }_{\mathrm{xx}}+e\left(t\right){\eta }_{\mathrm{xy}}){\eta }_x^2+e\left(t\right){\eta }_x{\eta }_y{\eta }_{\mathrm{xx}}){\varrho }^2{g}^{\mathrm{\prime }}{g}^{\mathrm{\prime \prime }}\\ & +((6a\left(t\right){\eta }_{\mathrm{xx}}+3b\left(t\right){\eta }_{\mathrm{xy}}){\eta }_x^2+3b\left(t\right){\eta }_x{\eta }_y{\eta }_{\mathrm{xx}})\varrho g^{\mathrm{\prime \prime \prime }}\\ & +((c\left(t\right){\phi }_{0_x}+e\left(t\right){\phi }_{0_y}){\eta }_x^2+(4a\left(t\right){\eta }_{\mathrm{xxx}}\\ & +3b\left(t\right){\eta }_{\mathrm{xxy}}+e\left(t\right){\phi }_{0_x}{\eta }_y+{\eta }_t){\eta }_x+b\left(t\right){\eta }_y{\eta }_{\mathrm{xxx}}\\ & +3{\eta }_{\mathrm{xx}}(a\left(t\right){\eta }_{\mathrm{xx}}+b\left(t\right){\eta }_{\mathrm{yx}}))\varrho g^{\mathrm{\prime \prime }}+{\phi }_{0_t}+a\left(t\right){\phi }_{0_{\mathrm{xxx}}}\\ & +b\left(t\right){\phi }_{0\mathrm{xxy}}+\frac{1}{2}c\left(t\right){\phi }_{0_x}^2+e\left(t\right){\phi }_{0_x}{\phi }_{0_y}\\ & ({\eta }_{\mathrm{xt}}+a\left(t\right){\eta }_{\mathrm{xxxx}}+b\left(t\right){\eta }_{\mathrm{xxxy}}\\ & +(c\left(t\right){\phi }_{0_x}+e\left(t\right){\phi }_{0_y}){\eta }_{\mathrm{xx}}\\ & +e\left(t\right){\phi }_{0_x}{\eta }_{\mathrm{xy}})\varrho g^{\mathrm{\prime }}=0.\end{array}\end{array}$(6) ), a nonlinear ordinary equation in η, equate all terms with an arbitrary function ${ϝ}_i(\eta ),$ $i=1\dots 7,$ multiplied by the coefficient of the most dispersive term $g^{\mathrm{\prime \prime \prime \prime }}$ (7) $\begin{array}{r}\begin{array}{rl}& (\frac{1}{2}c\left(t\right){\eta }_x+e\left(t\right){\eta }_y){\eta }_x^3{\varrho }^2\\ & ={ϝ}_1\left(\eta \right)(a\left(t\right){\eta }_x+b\left(t\right){\eta }_y){\eta }_x^3\varrho ,\\ & (\frac{1}{2}c\left(t\right){\eta }_{\mathrm{xx}}+e\left(t\right){\eta }_{\mathrm{yx}}){\eta }_{\mathrm{xx}}{\varrho }^2\\ & ={ϝ}_2\left(\eta \right)(a\left(t\right){\eta }_x+b\left(t\right){\eta }_y){\eta }_x^3\varrho ,\\ & ((c\left(t\right){\eta }_{\mathrm{xx}}+e\left(t\right){\eta }_{\mathrm{xy}}){\eta }_x^2+e\left(t\right){\eta }_x{\eta }_y{\eta }_{\mathrm{xx}}){\varrho }^2\\ & ={ϝ}_3\left(\eta \right)(a\left(t\right){\eta }_x+b\left(t\right){\eta }_y){\eta }_x^3\varrho ,\\ & ((6a\left(t\right){\eta }_{\mathrm{xx}}+3b\left(t\right){\eta }_{\mathrm{xy}}){\eta }_x^2+3b\left(t\right){\eta }_x{\eta }_y{\eta }_{\mathrm{xx}})\varrho \\ & ={ϝ}_4\left(\eta \right)(a\left(t\right){\eta }_x+b\left(t\right){\eta }_y){\eta }_x^3\varrho ,\\ & ((c\left(t\right){\phi }_{0_x}+e\left(t\right){\phi }_{0_y}){\eta }_x^2+(4a\left(t\right){\eta }_{\mathrm{xxx}}\\ & +3b\left(t\right){\eta }_{\mathrm{xxy}}+e\left(t\right){\phi }_{0_x}{\eta }_y+{\eta }_t){\eta }_x+b\left(t\right){\eta }_y{\eta }_{\mathrm{xxx}})\\ & +3{\eta }_{\mathrm{xx}}(a\left(t\right){\eta }_{\mathrm{xx}}+b\left(t\right){\eta }_{\mathrm{yx}}))\varrho ={ϝ}_5\left(\eta \right)(a\left(t\right){\eta }_x\\ & +b\left(t\right){\eta }_y){\eta }_x^3\varrho ,\\ & ({\eta }_{\mathrm{xt}}+a\left(t\right){\eta }_{\mathrm{xxxx}}+b\left(t\right){\eta }_{\mathrm{xxxy}}+(c\left(t\right){\phi }_{0_x}+e\left(t\right){\phi }_{0_y}){\eta }_{\mathrm{xx}}\\ & +e\left(t\right){\phi }_{0_x}{\eta }_{\mathrm{xy}})\varrho \\ & ={ϝ}_6\left(\eta \right)(a\left(t\right){\eta }_x+b\left(t\right){\eta }_y){\eta }_x^3\varrho ,\\ & {\phi }_{0_t}+a\left(t\right){\phi }_{0_{\mathrm{xxx}}}+b\left(t\right){\phi }_{0\mathrm{xxy}}\\ & +\frac{1}{2}c\left(t\right){\phi }_{0_x}^2+e\left(t\right){\phi }_{0_x}{\phi }_{0_y}={ϝ}_7\left(\eta \right)(a\left(t\right){\eta }_x\\ & +b\left(t\right){\eta }_y){\eta }_x^3\varrho .\end{array}\end{array}$(7) By solving system (7(7) $\begin{array}{r}\begin{array}{rl}& (\frac{1}{2}c\left(t\right){\eta }_x+e\left(t\right){\eta }_y){\eta }_x^3{\varrho }^2\\ & ={ϝ}_1\left(\eta \right)(a\left(t\right){\eta }_x+b\left(t\right){\eta }_y){\eta }_x^3\varrho ,\\ & (\frac{1}{2}c\left(t\right){\eta }_{\mathrm{xx}}+e\left(t\right){\eta }_{\mathrm{yx}}){\eta }_{\mathrm{xx}}{\varrho }^2\\ & ={ϝ}_2\left(\eta \right)(a\left(t\right){\eta }_x+b\left(t\right){\eta }_y){\eta }_x^3\varrho ,\\ & ((c\left(t\right){\eta }_{\mathrm{xx}}+e\left(t\right){\eta }_{\mathrm{xy}}){\eta }_x^2+e\left(t\right){\eta }_x{\eta }_y{\eta }_{\mathrm{xx}}){\varrho }^2\\ & ={ϝ}_3\left(\eta \right)(a\left(t\right){\eta }_x+b\left(t\right){\eta }_y){\eta }_x^3\varrho ,\\ & ((6a\left(t\right){\eta }_{\mathrm{xx}}+3b\left(t\right){\eta }_{\mathrm{xy}}){\eta }_x^2+3b\left(t\right){\eta }_x{\eta }_y{\eta }_{\mathrm{xx}})\varrho \\ & ={ϝ}_4\left(\eta \right)(a\left(t\right){\eta }_x+b\left(t\right){\eta }_y){\eta }_x^3\varrho ,\\ & ((c\left(t\right){\phi }_{0_x}+e\left(t\right){\phi }_{0_y}){\eta }_x^2+(4a\left(t\right){\eta }_{\mathrm{xxx}}\\ & +3b\left(t\right){\eta }_{\mathrm{xxy}}+e\left(t\right){\phi }_{0_x}{\eta }_y+{\eta }_t){\eta }_x+b\left(t\right){\eta }_y{\eta }_{\mathrm{xxx}})\\ & +3{\eta }_{\mathrm{xx}}(a\left(t\right){\eta }_{\mathrm{xx}}+b\left(t\right){\eta }_{\mathrm{yx}}))\varrho ={ϝ}_5\left(\eta \right)(a\left(t\right){\eta }_x\\ & +b\left(t\right){\eta }_y){\eta }_x^3\varrho ,\\ & ({\eta }_{\mathrm{xt}}+a\left(t\right){\eta }_{\mathrm{xxxx}}+b\left(t\right){\eta }_{\mathrm{xxxy}}+(c\left(t\right){\phi }_{0_x}+e\left(t\right){\phi }_{0_y}){\eta }_{\mathrm{xx}}\\ & +e\left(t\right){\phi }_{0_x}{\eta }_{\mathrm{xy}})\varrho \\ & ={ϝ}_6\left(\eta \right)(a\left(t\right){\eta }_x+b\left(t\right){\eta }_y){\eta }_x^3\varrho ,\\ & {\phi }_{0_t}+a\left(t\right){\phi }_{0_{\mathrm{xxx}}}+b\left(t\right){\phi }_{0\mathrm{xxy}}\\ & +\frac{1}{2}c\left(t\right){\phi }_{0_x}^2+e\left(t\right){\phi }_{0_x}{\phi }_{0_y}={ϝ}_7\left(\eta \right)(a\left(t\right){\eta }_x\\ & +b\left(t\right){\eta }_y){\eta }_x^3\varrho .\end{array}\end{array}$(7) ), we get the following values: (8) $\begin{array}{rl}{ϝ}_1\left(\eta \right)& =k_1,\\ {ϝ}_2\left(\eta \right)& ={ϝ}_3\left(\eta \right)={ϝ}_4\left(\eta \right)={ϝ}_6\left(\eta \right)=0,\\ {ϝ}_5\left(\eta \right)& =k_2,{ϝ}_7\left(\eta \right)=k_3,\end{array}$(8) (9) $\begin{array}{rl}\eta & =r_{1}x+r_{2}y+r_1^2(k_2-2\varrho k_3\sqrt{\frac{\varrho k_3}{k_1}})\\ & \times \int (r_{1}a\left(t\right)+r_{2}b\left(t\right))\mathrm{d}t,\end{array}$(9) (10) $\begin{array}{rl}c\left(t\right)& =\frac{2k_{1}a\left(t\right)}{\varrho },e\left(t\right)=\frac{k_{1}b\left(t\right)}{\varrho },\\ {\phi }_0(x,y,t)& =\sqrt{\frac{\varrho k_3}{k_1}}\varrho r_1(r_{1}x+r_{2}y)+\widetilde{{\phi }_0},\end{array}$(10) where $r_i,k_i,i=1,2,3$ and $\widetilde{{\phi }_0}$ are arbitrary constants. The vcBK equation reduces to (11) $\begin{array}{r}{k}_1{g}^{\mathrm{\prime \prime }2}+k_2{g}^{\mathrm{\prime \prime }}+g^{\mathrm{\prime \prime \prime \prime }}+k_3=0.\end{array}$(11) Put $g^{\mathrm{\prime }}=\upsilon (\eta ),$ to be easy to find the solutions of vcBK, from assumption (5(5) $\begin{array}{r}\phi (x,y,t)={\phi }_0(x,y,t)+\varrho \frac{\mathrm{\partial }g\left(\eta (x,y,t)\right)}{\mathrm{\partial }x}.\end{array}$(5) ), (12) $\begin{array}{r}\phi (x,y,t)=\varrho r_1(\upsilon (\eta )+\sqrt{\frac{\varrho k_3}{k_1}}(r_{1}x+r_{2}y))+\widetilde{{\phi }_0},\end{array}$(12) where $\upsilon (\eta )$ satisfies the nonlinear ordinary equation (13) $\begin{array}{r}{k}_1{\upsilon }^{\mathrm{\prime }2}+k_2{\upsilon }^{\mathrm{\prime }}+{\upsilon }^{\mathrm{\prime \prime \prime }}+k_3=0.\end{array}$(13) To solve the above equation we use the Riccati method [22, 32–34] as follows:

Assume $\upsilon (\eta )=\sum _{i=1}^n{A}_i{\psi }^i(\eta ),$ where n is determine by using the balance method again between the nonlinear term ${\upsilon }^{\mathrm{\prime }2}$ and the linear term ${\upsilon }^{\mathrm{\prime \prime \prime }}$, therefore n = 1, and (14) $\begin{array}{r}\upsilon (\eta )=A_0+A_1\psi \left(\eta \right),\end{array}$(14) where $\psi (\eta )$ satisfies the Riccati equation (15) $\begin{array}{r}{\psi }^{\mathrm{\prime }}\left(\eta \right)=r+\mathrm{p\psi }\left(\eta \right)+q{\psi }^2\left(\eta \right),\end{array}$(15) where r, p, q are arbitrary constants take different known values (for more details, see [22, 32–34]), then by substitution from (15(15) $\begin{array}{r}{\psi }^{\mathrm{\prime }}\left(\eta \right)=r+\mathrm{p\psi }\left(\eta \right)+q{\psi }^2\left(\eta \right),\end{array}$(15) ) into (13(13) $\begin{array}{r}{k}_1{\upsilon }^{\mathrm{\prime }2}+k_2{\upsilon }^{\mathrm{\prime }}+{\upsilon }^{\mathrm{\prime \prime \prime }}+k_3=0.\end{array}$(13) ) using Equation (14(14) $\begin{array}{r}\upsilon (\eta )=A_0+A_1\psi \left(\eta \right),\end{array}$(14) ), a polynomial in $\psi (\eta )$ is formed and by collecting and equating different coefficients of it with zero, an algebraic system appears by using Maple software the following relations between constants is obtained (16) $\begin{array}{r}{A}_1=\frac{-6q}{k_1},A_0=0,k_2=-p^2+4\mathrm{qr},k_3=0.\end{array}$(16) From [32], by using the values of r, p, q and the given values for the constants $A_0,A_1,A_2$ in Equations (14(14) $\begin{array}{r}\upsilon (\eta )=A_0+A_1\psi \left(\eta \right),\end{array}$(14) ) and (12(12) $\begin{array}{r}\phi (x,y,t)=\varrho r_1(\upsilon (\eta )+\sqrt{\frac{\varrho k_3}{k_1}}(r_{1}x+r_{2}y))+\widetilde{{\phi }_0},\end{array}$(12) ), the following new solutions for the vcBK equation are obtained: (17) $\begin{array}{rl}{\phi }_1& =\frac{3r_1\varrho }{k_1}(1+\tanh (\frac{1}{2}(r_{1}x\\ & +r_{2}y-r_1^2\int (r_{1}a\left(t\right)+r_{2}b\left(t\right))\mathrm{d}t)))+\widetilde{{\phi }_0},\end{array}$(17) (18) $\begin{array}{rl}{\phi }_2& =\frac{-3r_1\varrho }{k_1}(1-\coth (\frac{1}{2}(r_{1}x\\ & +r_{2}y-r_1^2\int (r_{1}a\left(t\right)+r_{2}b\left(t\right))\mathrm{d}t)))+\widetilde{{\phi }_0},\end{array}$(18) (19) $\begin{array}{rl}{\phi }_3& =\frac{3r_1\varrho }{k_1}(\coth (r_{1}x+r_{2}y\\ & -r_1^2\int (r_{1}a\left(t\right)+r_{2}b\left(t\right))\mathrm{d}t)\\ & \pm \mathrm{csch}(r_{1}x+r_{2}y\\ & -r_1^2\int (r_{1}a\left(t\right)+r_{2}b\left(t\right))\mathrm{d}t))+\widetilde{{\phi }_0},\end{array}$(19) (20) $\begin{array}{rl}{\phi }_4& =\frac{3r_1\varrho }{k_1}(\tanh (r_{1}x+r_{2}y\\ & -r_1^2\int (r_{1}a\left(t\right)+r_{2}b\left(t\right))\mathrm{d}t)\\ & \pm i\mathrm{sech}(r_{1}x+r_{2}y\\ & -r_1^2\int (r_{1}a\left(t\right)+r_{2}b\left(t\right))\mathrm{d}t))+\widetilde{{\phi }_0},\end{array}$(20) (21) $\begin{array}{rl}{\phi }_5& =\frac{\mp 3r_1\varrho }{k_1}(\sec (r_{1}x+r_{2}y\\ & -r_1^2\int (r_{1}a\left(t\right)+r_{2}b\left(t\right))\mathrm{d}t)\\ & \pm \tan (r_{1}x+r_{2}y\\ & -r_1^2\int (r_{1}a\left(t\right)+r_{2}b\left(t\right))\mathrm{d}t))+\widetilde{{\phi }_0},\end{array}$(21) (22) $\begin{array}{rl}{\phi }_6& =\frac{\pm 3r_1\varrho }{k_1}(\csc (r_{1}x+r_{2}y-r_1^2\int (r_{1}a\left(t\right)\\ & +r_{2}b\left(t\right))\mathrm{d}t)\\ & \mp \cot (r_{1}x+r_{2}y-r_1^2\int (r_{1}a\left(t\right)\\ & +r_{2}b\left(t\right))\mathrm{d}t))+\widetilde{{\phi }_0},\end{array}$(22) (23) $\begin{array}{rl}{\phi }_7& =\frac{6r_1\varrho }{k_1}\tanh (r_{1}x+r_{2}y\\ & -4r_1^2\int (r_{1}a\left(t\right)+r_{2}b\left(t\right))\mathrm{d}t+\widetilde{{\phi }_0},\end{array}$(23) (24) $\begin{array}{rl}{\phi }_8& =\frac{6r_1\varrho }{k_1}\coth (r_{1}x+r_{2}y\\ & -4r_1^2\int (r_{1}a\left(t\right)+r_{2}b\left(t\right))\mathrm{d}t+\widetilde{{\phi }_0},\end{array}$(24) (25) $\begin{array}{rl}{\phi }_9& =\frac{-6r_1\varrho }{k_1}\tan (r_{1}x+r_{2}y\\ & +4r_1^2\int (r_{1}a\left(t\right)+r_{2}b\left(t\right))\mathrm{d}t+\widetilde{{\phi }_0},\end{array}$(25) (26) $\begin{array}{rl}{\phi }_{10}& =\frac{6r_1\varrho }{k_1}\cot (r_{1}x+r_{2}y\\ & +4r_1^2\int (r_{1}a\left(t\right)+r_{2}b\left(t\right))\mathrm{d}t+\widetilde{{\phi }_0}.\end{array}$(26)

3. The HB method and Bäklund transformation

In this section, we are going to find Bäcklund transformation for the vcBK equation by using the same assumption (5(5) $\begin{array}{r}\phi (x,y,t)={\phi }_0(x,y,t)+\varrho \frac{\mathrm{\partial }g\left(\eta (x,y,t)\right)}{\mathrm{\partial }x}.\end{array}$(5) ), but in another way, by rearrange terms for the η derivatives (27) $\begin{array}{rl}& (a\left(t\right)g^{\mathrm{\prime \prime \prime \prime }}+\frac{1}{2}c\left(t\right)g^{\mathrm{\prime \prime }2}\varrho )\varrho {\eta }_x^4\\ & +(b\left(t\right)g^{\mathrm{\prime \prime \prime \prime }}+e\left(t\right)g^{\mathrm{\prime \prime }2}\varrho )\varrho {\eta }_y{\eta }_x^3\\ & +(6a\left(t\right)g^{\mathrm{\prime \prime \prime }}+c\left(t\right)\varrho g^{\mathrm{\prime }}{g}^{\mathrm{\prime \prime }})\varrho {\eta }_{\mathrm{xx}}{\eta }_x^2\\ & +(3b\left(t\right)g^{\mathrm{\prime \prime \prime }}+e\left(t\right)\varrho g^{\mathrm{\prime }}{g}^{\mathrm{\prime \prime }})\varrho {\eta }_{\mathrm{xy}}{\eta }_x^2\\ & +(c\left(t\right){\phi }_{0_x}+e\left(t\right){\phi }_{0_y})\varrho g^{\mathrm{\prime \prime }}{\eta }_x^2+4a\left(t\right)\varrho g^{\mathrm{\prime \prime }}{\eta }_x{\eta }_{\mathrm{xxx}}\\ & +(3b\left(t\right)g^{\mathrm{\prime \prime \prime }}+e\left(t\right)\varrho g^{\mathrm{\prime }}{g}^{\mathrm{\prime \prime }})\varrho {\eta }_x{\eta }_y{\eta }_{\mathrm{xx}}\\ & +3\mathrm{\varrho b}\left(t\right)g^{\mathrm{\prime \prime }}{\eta }_x{\eta }_{\mathrm{xxy}}+e\left(t\right)\varrho {\phi }_{0_x}{g}^{\mathrm{\prime \prime }}{\eta }_x{\eta }_y+\varrho g^{\mathrm{\prime \prime }}{\eta }_x{\eta }_t\\ & +\varrho g^{\mathrm{\prime }}{\eta }_{\mathrm{xt}}+(3a\left(t\right)g^{\mathrm{\prime \prime }}+\frac{1}{2}\mathrm{\varrho c}\left(t\right)g^{\mathrm{\prime }2})\varrho {\eta }_{\mathrm{xx}}^2\\ & +a\left(t\right)\varrho g^{\mathrm{\prime }}{\eta }_{\mathrm{xxxx}}+(3b\left(t\right)g^{\mathrm{\prime \prime }}+e\left(t\right)\varrho g^{\mathrm{\prime }2}){\eta }_{\mathrm{yx}}{\eta }_{\mathrm{xx}}\\ & +b\left(t\right)\varrho g^{\mathrm{\prime \prime }}{\eta }_y{\eta }_{\mathrm{xxx}}+b\left(t\right)\varrho g^{\mathrm{\prime }}{\eta }_{\mathrm{xxxy}}\\ & +(c\left(t\right){\phi }_{0_x}+e\left(t\right){\phi }_{0_y})\varrho g^{\mathrm{\prime }}{\eta }_{\mathrm{xx}}+e\left(t\right){\phi }_{0_x}\varrho g^{\mathrm{\prime }}{\eta }_{\mathrm{xy}}\\ & +{\phi }_{0_t}+a\left(t\right){\phi }_{0_{\mathrm{xxx}}}+b\left(t\right){\phi }_{0\mathrm{xxy}}\\ & +\frac{1}{2}c\left(t\right){\phi }_{0_x}^2+e\left(t\right){\phi }_{0_x}{\phi }_{0_y}=0.\end{array}$(27) To solve Equation (27(27) $\begin{array}{rl}& (a\left(t\right)g^{\mathrm{\prime \prime \prime \prime }}+\frac{1}{2}c\left(t\right)g^{\mathrm{\prime \prime }2}\varrho )\varrho {\eta }_x^4\\ & +(b\left(t\right)g^{\mathrm{\prime \prime \prime \prime }}+e\left(t\right)g^{\mathrm{\prime \prime }2}\varrho )\varrho {\eta }_y{\eta }_x^3\\ & +(6a\left(t\right)g^{\mathrm{\prime \prime \prime }}+c\left(t\right)\varrho g^{\mathrm{\prime }}{g}^{\mathrm{\prime \prime }})\varrho {\eta }_{\mathrm{xx}}{\eta }_x^2\\ & +(3b\left(t\right)g^{\mathrm{\prime \prime \prime }}+e\left(t\right)\varrho g^{\mathrm{\prime }}{g}^{\mathrm{\prime \prime }})\varrho {\eta }_{\mathrm{xy}}{\eta }_x^2\\ & +(c\left(t\right){\phi }_{0_x}+e\left(t\right){\phi }_{0_y})\varrho g^{\mathrm{\prime \prime }}{\eta }_x^2+4a\left(t\right)\varrho g^{\mathrm{\prime \prime }}{\eta }_x{\eta }_{\mathrm{xxx}}\\ & +(3b\left(t\right)g^{\mathrm{\prime \prime \prime }}+e\left(t\right)\varrho g^{\mathrm{\prime }}{g}^{\mathrm{\prime \prime }})\varrho {\eta }_x{\eta }_y{\eta }_{\mathrm{xx}}\\ & +3\mathrm{\varrho b}\left(t\right)g^{\mathrm{\prime \prime }}{\eta }_x{\eta }_{\mathrm{xxy}}+e\left(t\right)\varrho {\phi }_{0_x}{g}^{\mathrm{\prime \prime }}{\eta }_x{\eta }_y+\varrho g^{\mathrm{\prime \prime }}{\eta }_x{\eta }_t\\ & +\varrho g^{\mathrm{\prime }}{\eta }_{\mathrm{xt}}+(3a\left(t\right)g^{\mathrm{\prime \prime }}+\frac{1}{2}\mathrm{\varrho c}\left(t\right)g^{\mathrm{\prime }2})\varrho {\eta }_{\mathrm{xx}}^2\\ & +a\left(t\right)\varrho g^{\mathrm{\prime }}{\eta }_{\mathrm{xxxx}}+(3b\left(t\right)g^{\mathrm{\prime \prime }}+e\left(t\right)\varrho g^{\mathrm{\prime }2}){\eta }_{\mathrm{yx}}{\eta }_{\mathrm{xx}}\\ & +b\left(t\right)\varrho g^{\mathrm{\prime \prime }}{\eta }_y{\eta }_{\mathrm{xxx}}+b\left(t\right)\varrho g^{\mathrm{\prime }}{\eta }_{\mathrm{xxxy}}\\ & +(c\left(t\right){\phi }_{0_x}+e\left(t\right){\phi }_{0_y})\varrho g^{\mathrm{\prime }}{\eta }_{\mathrm{xx}}+e\left(t\right){\phi }_{0_x}\varrho g^{\mathrm{\prime }}{\eta }_{\mathrm{xy}}\\ & +{\phi }_{0_t}+a\left(t\right){\phi }_{0_{\mathrm{xxx}}}+b\left(t\right){\phi }_{0\mathrm{xxy}}\\ & +\frac{1}{2}c\left(t\right){\phi }_{0_x}^2+e\left(t\right){\phi }_{0_x}{\phi }_{0_y}=0.\end{array}$(27) ), put the coefficients of ${\eta }_x^4$ equal zero (28) $\begin{array}{r}a\left(t\right)g^{\mathrm{\prime \prime \prime \prime }}+\frac{1}{2}c\left(t\right)g^{\mathrm{\prime \prime }2}\varrho =0.\end{array}$(28) Equation (28(28) $\begin{array}{r}a\left(t\right)g^{\mathrm{\prime \prime \prime \prime }}+\frac{1}{2}c\left(t\right)g^{\mathrm{\prime \prime }2}\varrho =0.\end{array}$(28) ) has the following solution: (29) $\begin{array}{r}g=\frac{12a\left(t\right)}{\mathrm{\varrho c}\left(t\right)}\ln \left(\eta \right).\end{array}$(29) By back substitution from Equation (29(29) $\begin{array}{r}g=\frac{12a\left(t\right)}{\mathrm{\varrho c}\left(t\right)}\ln \left(\eta \right).\end{array}$(29) ) in Equation (27(27) $\begin{array}{rl}& (a\left(t\right)g^{\mathrm{\prime \prime \prime \prime }}+\frac{1}{2}c\left(t\right)g^{\mathrm{\prime \prime }2}\varrho )\varrho {\eta }_x^4\\ & +(b\left(t\right)g^{\mathrm{\prime \prime \prime \prime }}+e\left(t\right)g^{\mathrm{\prime \prime }2}\varrho )\varrho {\eta }_y{\eta }_x^3\\ & +(6a\left(t\right)g^{\mathrm{\prime \prime \prime }}+c\left(t\right)\varrho g^{\mathrm{\prime }}{g}^{\mathrm{\prime \prime }})\varrho {\eta }_{\mathrm{xx}}{\eta }_x^2\\ & +(3b\left(t\right)g^{\mathrm{\prime \prime \prime }}+e\left(t\right)\varrho g^{\mathrm{\prime }}{g}^{\mathrm{\prime \prime }})\varrho {\eta }_{\mathrm{xy}}{\eta }_x^2\\ & +(c\left(t\right){\phi }_{0_x}+e\left(t\right){\phi }_{0_y})\varrho g^{\mathrm{\prime \prime }}{\eta }_x^2+4a\left(t\right)\varrho g^{\mathrm{\prime \prime }}{\eta }_x{\eta }_{\mathrm{xxx}}\\ & +(3b\left(t\right)g^{\mathrm{\prime \prime \prime }}+e\left(t\right)\varrho g^{\mathrm{\prime }}{g}^{\mathrm{\prime \prime }})\varrho {\eta }_x{\eta }_y{\eta }_{\mathrm{xx}}\\ & +3\mathrm{\varrho b}\left(t\right)g^{\mathrm{\prime \prime }}{\eta }_x{\eta }_{\mathrm{xxy}}+e\left(t\right)\varrho {\phi }_{0_x}{g}^{\mathrm{\prime \prime }}{\eta }_x{\eta }_y+\varrho g^{\mathrm{\prime \prime }}{\eta }_x{\eta }_t\\ & +\varrho g^{\mathrm{\prime }}{\eta }_{\mathrm{xt}}+(3a\left(t\right)g^{\mathrm{\prime \prime }}+\frac{1}{2}\mathrm{\varrho c}\left(t\right)g^{\mathrm{\prime }2})\varrho {\eta }_{\mathrm{xx}}^2\\ & +a\left(t\right)\varrho g^{\mathrm{\prime }}{\eta }_{\mathrm{xxxx}}+(3b\left(t\right)g^{\mathrm{\prime \prime }}+e\left(t\right)\varrho g^{\mathrm{\prime }2}){\eta }_{\mathrm{yx}}{\eta }_{\mathrm{xx}}\\ & +b\left(t\right)\varrho g^{\mathrm{\prime \prime }}{\eta }_y{\eta }_{\mathrm{xxx}}+b\left(t\right)\varrho g^{\mathrm{\prime }}{\eta }_{\mathrm{xxxy}}\\ & +(c\left(t\right){\phi }_{0_x}+e\left(t\right){\phi }_{0_y})\varrho g^{\mathrm{\prime }}{\eta }_{\mathrm{xx}}+e\left(t\right){\phi }_{0_x}\varrho g^{\mathrm{\prime }}{\eta }_{\mathrm{xy}}\\ & +{\phi }_{0_t}+a\left(t\right){\phi }_{0_{\mathrm{xxx}}}+b\left(t\right){\phi }_{0\mathrm{xxy}}\\ & +\frac{1}{2}c\left(t\right){\phi }_{0_x}^2+e\left(t\right){\phi }_{0_x}{\phi }_{0_y}=0.\end{array}$(27) ), we obtain the following integrability conditions: (30) $\begin{array}{r}a(t)=\mathrm{\varrho c}\left(t\right),b\left(t\right)=2\mathrm{\varrho e}\left(t\right),{\phi }_0=\mathrm{constant}.\end{array}$(30) The required Bäcklund transformation is (31) $\begin{array}{r}\phi (x,y,t)={\phi }_0+\frac{12\varrho {\eta }_x}{\eta (x,y,t)}.\end{array}$(31) To find N-soliton solutions for the vcBK equation, we should solve Equation (27(27) $\begin{array}{rl}& (a\left(t\right)g^{\mathrm{\prime \prime \prime \prime }}+\frac{1}{2}c\left(t\right)g^{\mathrm{\prime \prime }2}\varrho )\varrho {\eta }_x^4\\ & +(b\left(t\right)g^{\mathrm{\prime \prime \prime \prime }}+e\left(t\right)g^{\mathrm{\prime \prime }2}\varrho )\varrho {\eta }_y{\eta }_x^3\\ & +(6a\left(t\right)g^{\mathrm{\prime \prime \prime }}+c\left(t\right)\varrho g^{\mathrm{\prime }}{g}^{\mathrm{\prime \prime }})\varrho {\eta }_{\mathrm{xx}}{\eta }_x^2\\ & +(3b\left(t\right)g^{\mathrm{\prime \prime \prime }}+e\left(t\right)\varrho g^{\mathrm{\prime }}{g}^{\mathrm{\prime \prime }})\varrho {\eta }_{\mathrm{xy}}{\eta }_x^2\\ & +(c\left(t\right){\phi }_{0_x}+e\left(t\right){\phi }_{0_y})\varrho g^{\mathrm{\prime \prime }}{\eta }_x^2+4a\left(t\right)\varrho g^{\mathrm{\prime \prime }}{\eta }_x{\eta }_{\mathrm{xxx}}\\ & +(3b\left(t\right)g^{\mathrm{\prime \prime \prime }}+e\left(t\right)\varrho g^{\mathrm{\prime }}{g}^{\mathrm{\prime \prime }})\varrho {\eta }_x{\eta }_y{\eta }_{\mathrm{xx}}\\ & +3\mathrm{\varrho b}\left(t\right)g^{\mathrm{\prime \prime }}{\eta }_x{\eta }_{\mathrm{xxy}}+e\left(t\right)\varrho {\phi }_{0_x}{g}^{\mathrm{\prime \prime }}{\eta }_x{\eta }_y+\varrho g^{\mathrm{\prime \prime }}{\eta }_x{\eta }_t\\ & +\varrho g^{\mathrm{\prime }}{\eta }_{\mathrm{xt}}+(3a\left(t\right)g^{\mathrm{\prime \prime }}+\frac{1}{2}\mathrm{\varrho c}\left(t\right)g^{\mathrm{\prime }2})\varrho {\eta }_{\mathrm{xx}}^2\\ & +a\left(t\right)\varrho g^{\mathrm{\prime }}{\eta }_{\mathrm{xxxx}}+(3b\left(t\right)g^{\mathrm{\prime \prime }}+e\left(t\right)\varrho g^{\mathrm{\prime }2}){\eta }_{\mathrm{yx}}{\eta }_{\mathrm{xx}}\\ & +b\left(t\right)\varrho g^{\mathrm{\prime \prime }}{\eta }_y{\eta }_{\mathrm{xxx}}+b\left(t\right)\varrho g^{\mathrm{\prime }}{\eta }_{\mathrm{xxxy}}\\ & +(c\left(t\right){\phi }_{0_x}+e\left(t\right){\phi }_{0_y})\varrho g^{\mathrm{\prime }}{\eta }_{\mathrm{xx}}+e\left(t\right){\phi }_{0_x}\varrho g^{\mathrm{\prime }}{\eta }_{\mathrm{xy}}\\ & +{\phi }_{0_t}+a\left(t\right){\phi }_{0_{\mathrm{xxx}}}+b\left(t\right){\phi }_{0\mathrm{xxy}}\\ & +\frac{1}{2}c\left(t\right){\phi }_{0_x}^2+e\left(t\right){\phi }_{0_x}{\phi }_{0_y}=0.\end{array}$(27) ) after conditions in Equation (31(31) $\begin{array}{r}\phi (x,y,t)={\phi }_0+\frac{12\varrho {\eta }_x}{\eta (x,y,t)}.\end{array}$(31) ), which becomes (32) $\begin{array}{rl}& ({\eta }_{\mathrm{xt}}+(c\left(t\right){\eta }_{\mathrm{xxxx}}+2e(t){\eta }_{\mathrm{xxxy}})\varrho )\eta \\ & -2\varrho (2c(t){\eta }_x{\eta }_{\mathrm{xxx}}+3e\left(t\right){\eta }_x{\eta }_{\mathrm{xxy}})\\ & +3\varrho (c\left(t\right){\eta }_{\mathrm{xx}}^2+2e(t){\eta }_{\mathrm{yx}}{\eta }_{\mathrm{xx}})\\ & -{\eta }_x{\eta }_t-2\mathrm{\varrho e}(t){\eta }_y{\eta }_{\mathrm{xxx}}=0.\end{array}$(32) For one-soliton solution assume $\eta =1+\exp (p_{1}x+q_{1}y-\psi (t)+{\alpha }_1),$ then the wave number $\psi (t)=\varrho p_1^2\int (c(t)p_1+2e(t)q_1)\mathrm{dt},$ and by back substitution in (32(32) $\begin{array}{rl}& ({\eta }_{\mathrm{xt}}+(c\left(t\right){\eta }_{\mathrm{xxxx}}+2e(t){\eta }_{\mathrm{xxxy}})\varrho )\eta \\ & -2\varrho (2c(t){\eta }_x{\eta }_{\mathrm{xxx}}+3e\left(t\right){\eta }_x{\eta }_{\mathrm{xxy}})\\ & +3\varrho (c\left(t\right){\eta }_{\mathrm{xx}}^2+2e(t){\eta }_{\mathrm{yx}}{\eta }_{\mathrm{xx}})\\ & -{\eta }_x{\eta }_t-2\mathrm{\varrho e}(t){\eta }_y{\eta }_{\mathrm{xxx}}=0.\end{array}$(32) ), the following one-soliton solution for the vcBK equation is given as (33) $\begin{array}{rl}& {\phi }_{11}(x,y,t)\\ & =\frac{12\varrho p_1{\mathrm{e}}^{(p_{1}x+q_{1}y-\varrho p_1^2\int (c\left(t\right)p_1+2e\left(t\right)q_1)\mathrm{d}t+{\alpha }_1)}}{1+{\mathrm{e}}^{(p_{1}x+q_{1}y-\varrho p_1^2\int (c\left(t\right)p_1+2e\left(t\right)q_1)\mathrm{d}t+{\alpha }_1)}}+{\phi }_0.\end{array}$(33) For two-soliton solutions put $\eta =1+\sum _{i=1}^{i=2}{p}_i{\mathrm{e}}^{(p_i(x+y)-\varrho p_i^3\int (c(t)+2e(t))\mathrm{d}t+{\alpha }_i)}+s_{12}{\mathrm{e}}^{((p_1+p_2)(x+y)-\varrho (p_1^3}{}^{+p_2^3)\int (c(t)+2e(t))\mathrm{dt}+{\alpha }_1+{\alpha }_2)}$ in Equation (32(32) $\begin{array}{rl}& ({\eta }_{\mathrm{xt}}+(c\left(t\right){\eta }_{\mathrm{xxxx}}+2e(t){\eta }_{\mathrm{xxxy}})\varrho )\eta \\ & -2\varrho (2c(t){\eta }_x{\eta }_{\mathrm{xxx}}+3e\left(t\right){\eta }_x{\eta }_{\mathrm{xxy}})\\ & +3\varrho (c\left(t\right){\eta }_{\mathrm{xx}}^2+2e(t){\eta }_{\mathrm{yx}}{\eta }_{\mathrm{xx}})\\ & -{\eta }_x{\eta }_t-2\mathrm{\varrho e}(t){\eta }_y{\eta }_{\mathrm{xxx}}=0.\end{array}$(32) ), the constant $s_{12}=\frac{(p_1-p_2{)}^2}{(p_1+p_2{)}^2},$ is obtained with conditions $q_1=p_1$ and $q_2=p_2,$ so that the following two soliton solution for Equation (3(3) $\begin{array}{r}{\phi }_t+a\left(t\right){\phi }_{\mathrm{xxx}}+b\left(t\right){\phi }_{\mathrm{xxy}}+\frac{c\left(t\right)}{2}{\phi }_x^2+e\left(t\right){\phi }_x{\phi }_y=0.\end{array}$(3) ) is

(34) $\begin{array}{r}{\phi }_{12}=\frac{\begin{array}{c}12\varrho (\sum _{i=1}^{i=2}{p}_i{\mathrm{e}}^{(p_i(x+y)-\varrho p_i^3\int (c\left(t\right)+2e\left(t\right))\mathrm{d}t+{\alpha }_i)}\\ +\frac{{(p_1-p_2)}^2}{(p_1+p_2)}{\mathrm{e}}^{((p_1+p_2)(x+y)-\varrho (p_1^3+p_2^3)\int (c\left(t\right)+2e\left(t\right))\mathrm{d}t+{\alpha }_1+{\alpha }_2)})\end{array}}{\begin{array}{c}1+\sum _{i=1}^{i=2}{\mathrm{e}}^{(p_i(x+y)-\varrho p_i^3\int (c\left(t\right)+2e\left(t\right))\mathrm{d}t+{\alpha }_i)}\\ +\frac{{(p_1-p_2)}^2}{{(p_1+p_2)}^2}{\mathrm{e}}^{((p_1+p_2)(x+y)-\varrho (p_1^3+p_2^3)\int (c\left(t\right)+2e\left(t\right))\mathrm{d}t+{\alpha }_1+{\alpha }_2)}\end{array}}+{\phi }_0.\end{array}$(34)

By the same manner, we obtain the N-soliton solution with ${\chi }_i=(p_i(x+y)-\varrho p_i^3\int (c(t)+2e(t))\mathrm{d}t+{\alpha }_i)$ for simplicity (35) $\begin{array}{r}{\phi }_N=\frac{\begin{array}{c}12\varrho (\sum _{i=1}^{i=N}{p}_i{\mathrm{e}}^{{\chi }_i}+\sum _{j=1}^{j=N-1}\\ \sum _{k=j+1}^{k=N}\frac{{(p_j-p_k)}^2}{(p_j+p_k)}\\ {\mathrm{e}}^{{\chi }_j+{\chi }_k}+\prod _{1\le j<k\le N}\frac{{(p_j-p_k)}^2}{{(p_j+p_k)}^2}\\ \left(\sum _{i=1}^{i=N}{p}_i\right){\mathrm{e}}^{\sum _{i=1}^{i=N}{\chi }_i})\end{array}}{\begin{array}{c}1+\sum _{i=1}^{i=N}{\mathrm{e}}^{{\chi }_i}+\sum _{j=1}^{j=N-1}\\ \sum _{k=j+1}^{k=N}\frac{{(p_j-p_k)}^2}{{(p_j+p_k)}^2}{\mathrm{e}}^{{\chi }_j+{\chi }_k}\\ +\prod _{1\le j<k\le N}\frac{{(p_j-p_k)}^2}{{(p_j+p_k)}^2}{\mathrm{e}}^{\sum _{i=1}^{i=N}{\chi }_i}\end{array}}+{\phi }_0.\end{array}$(35)

4. Results and discussion

Solitons or solitary waves are an important phenomena in many fields of science like oceans, optical fibres, material physics, biology, etc. Solitary waves appear from the balance between nonlinear and dispersive effects in the medium. Therefore, in this section we are going to discuss the motivation of our results by plotting the one and two solitons given by Equations (33(33) $\begin{array}{rl}& {\phi }_{11}(x,y,t)\\ & =\frac{12\varrho p_1{\mathrm{e}}^{(p_{1}x+q_{1}y-\varrho p_1^2\int (c\left(t\right)p_1+2e\left(t\right)q_1)\mathrm{d}t+{\alpha }_1)}}{1+{\mathrm{e}}^{(p_{1}x+q_{1}y-\varrho p_1^2\int (c\left(t\right)p_1+2e\left(t\right)q_1)\mathrm{d}t+{\alpha }_1)}}+{\phi }_0.\end{array}$(33) ) and (34(34) $\begin{array}{r}{\phi }_{12}=\frac{\begin{array}{c}12\varrho (\sum _{i=1}^{i=2}{p}_i{\mathrm{e}}^{(p_i(x+y)-\varrho p_i^3\int (c\left(t\right)+2e\left(t\right))\mathrm{d}t+{\alpha }_i)}\\ +\frac{{(p_1-p_2)}^2}{(p_1+p_2)}{\mathrm{e}}^{((p_1+p_2)(x+y)-\varrho (p_1^3+p_2^3)\int (c\left(t\right)+2e\left(t\right))\mathrm{d}t+{\alpha }_1+{\alpha }_2)})\end{array}}{\begin{array}{c}1+\sum _{i=1}^{i=2}{\mathrm{e}}^{(p_i(x+y)-\varrho p_i^3\int (c\left(t\right)+2e\left(t\right))\mathrm{d}t+{\alpha }_i)}\\ +\frac{{(p_1-p_2)}^2}{{(p_1+p_2)}^2}{\mathrm{e}}^{((p_1+p_2)(x+y)-\varrho (p_1^3+p_2^3)\int (c\left(t\right)+2e\left(t\right))\mathrm{d}t+{\alpha }_1+{\alpha }_2)}\end{array}}+{\phi }_0.\end{array}$(34) ) for different values of variable coefficients to see how it can affect on the soliton propagation  in both three and two dimensional as follows:

Figure 1 represents the one-soliton solution ${\phi }_{11}$ and the two soliton wave solution ${\phi }_{12}$ given by Equations (33(33) $\begin{array}{rl}& {\phi }_{11}(x,y,t)\\ & =\frac{12\varrho p_1{\mathrm{e}}^{(p_{1}x+q_{1}y-\varrho p_1^2\int (c\left(t\right)p_1+2e\left(t\right)q_1)\mathrm{d}t+{\alpha }_1)}}{1+{\mathrm{e}}^{(p_{1}x+q_{1}y-\varrho p_1^2\int (c\left(t\right)p_1+2e\left(t\right)q_1)\mathrm{d}t+{\alpha }_1)}}+{\phi }_0.\end{array}$(33) ) and (34(34) $\begin{array}{r}{\phi }_{12}=\frac{\begin{array}{c}12\varrho (\sum _{i=1}^{i=2}{p}_i{\mathrm{e}}^{(p_i(x+y)-\varrho p_i^3\int (c\left(t\right)+2e\left(t\right))\mathrm{d}t+{\alpha }_i)}\\ +\frac{{(p_1-p_2)}^2}{(p_1+p_2)}{\mathrm{e}}^{((p_1+p_2)(x+y)-\varrho (p_1^3+p_2^3)\int (c\left(t\right)+2e\left(t\right))\mathrm{d}t+{\alpha }_1+{\alpha }_2)})\end{array}}{\begin{array}{c}1+\sum _{i=1}^{i=2}{\mathrm{e}}^{(p_i(x+y)-\varrho p_i^3\int (c\left(t\right)+2e\left(t\right))\mathrm{d}t+{\alpha }_i)}\\ +\frac{{(p_1-p_2)}^2}{{(p_1+p_2)}^2}{\mathrm{e}}^{((p_1+p_2)(x+y)-\varrho (p_1^3+p_2^3)\int (c\left(t\right)+2e\left(t\right))\mathrm{d}t+{\alpha }_1+{\alpha }_2)}\end{array}}+{\phi }_0.\end{array}$(34) ) in both 3D and 2D with fixed $p_1=2,\varrho ={\alpha }_1=q_1=1,{\phi }_0=y=0$ where the variable functions are taken as constants $c(t)=1,e(t)=\mathrm{0.5.}$

Figure 1. (a) The one soliton wave solution, (b) the two soliton wave solution, (c) the 2D plot of the one soliton wave and (d) the 2D plot of the two soliton wave.

Here is Figure 2 for the one and two soliton solutions with fixed parameters $p_1=\varrho ={\alpha }_1={\alpha }_2=1,p_2=2,{\phi }_0=y=0$ and the variable coefficients are linear functions in t, $c(t)=e(t)=t.$

Figure 2. (a) The kink soliton wave takes a parabolic, (b) the parabolic two soliton waves, (c) the 2D plot for the parabolic one soliton wave and (d) the 2D shape of the parabolic two soliton waves.

In Figure 3, the one and two solitons plotted with respect to the periodic functions c(t)=sin(t) and e(t)=cos(t), where the constants are $p_1=\varrho ={\alpha }_1={\alpha }_2=1,p_2=2,{\phi }_0=y=0.$

Figure 3. (a) The periodic one soliton wave ${\phi }_{11}$, (b) the periodic two solitons wave ${\phi }_{12}$, (c) the 2D plot of the periodic soliton wave ${\phi }_{11}$ and (d) the 2D plot of the two solitons ${\phi }_{12}$.

In all figures, the constants $p_1=\varrho ={\alpha }_1={\alpha }_2=1,p_2=2,{\phi }_0=y=0$ are fixed, but the variable coefficients $c(t)$ and $e(t)$ are changed as: constants $c(t)=1$ and $e(t)=0.5$ in Figure 1, so that the one soliton solution ${\phi }_{11}$ in Figure 1(a) as 3D and Figure 1(c) as 2D is like a kink-type soliton also the two solitons in Figure 1(b,d) are of kink type. In Figure 2, the variable $c(t)=e(t)=t$ are linear functions in t and the soliton propagation is affected by the variable speed $\int (c(t)+2e(t))\mathrm{dt}=\int 3\mathrm{tdt}=\frac{3}{2}{t}^2$, therefore it's shape like a parabola for both one soliton wave in Figures 2(a,c) and two solitons in Figure 3(b,d). Finally, in Figure 3, the variables $c(t)=\sin (t)$, $e(t)=\cos (t)$ so that the one soliton and two solitons like a periodic wave.

5. Conclusion

In this paper, the balance technique was used three times, in the second section, we have used the balance method connected with the direct similarity reduction method to reduce the vcBK into a nonlinear ordinary differential equation (NODE), then we have used the balance method again in the Riccati equation method as a technique to solve the reduced NODE, according to that many kink, soliton, periodic wave types of solutions were obtained. In Section 3, the balance procedure is used to drive Bä cklund transformation for the vcBK equation, as a result one soliton, two solitons and n-soliton solutions were obtained. Some concluding remarks about the new contributions of this paper are as follows:

1. The similarity transformation given by Equation (9(9) $\begin{array}{rl}\eta & =r_{1}x+r_{2}y+r_1^2(k_2-2\varrho k_3\sqrt{\frac{\varrho k_3}{k_1}})\\ & \times \int (r_{1}a\left(t\right)+r_{2}b\left(t\right))\mathrm{d}t,\end{array}$(9) ) is new and general than other similarity transformations obtained by Lie groups in [29–31].

2. The solitary and periodic wave solutions obtained in the second section from ${\phi }_1$ to ${\phi }_{10}$ are novel and not obtained before [27–31].

3. The obtained N-soliton solutions are more general than the one, two, and three solitons obtained in [27, 28].

4. The graphs given in Section 4 reflect the effect of the variable coefficients $c(t)$ and $e(t)$ on the propagation of the one and two soliton wave solutions.

5. The application of the balance technique to obtain N-soliton and periodic wave solutions makes the technique applicable for other NPDEs with variable coefficients and it was easy and straightforward.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The authors would like to thank the Deanship of Scientific Research, Majmaah University, Saudi Arabia, for funding this work under project no. R-2023-159.