1. Introduction
Fluid mechanics has been regarded as the study of the fundamental mechanisms and force of liquids, plasmas and gases, with applications in astrophysics, meteorology, oceanography and biomedical engineering [1–4]. Methods have been introduced to solve the nonlinear evolution equations, including the Bäcklund transformation, Darboux transformation and the Lie symmetry approach [5–10].
A (2+1)-dimensional variable-coefficient modified dispersive water-wave system in fluid mechanics has been constructed as [11]1 ) have been given via the bifunction method with the exponential form solution [11].
$\begin{eqnarray}\begin{array}{l}{u}_{{ty}}+\alpha (t){u}_{{xxy}}-2\alpha (t){v}_{{xx}}\\ \,-\beta (t){{uu}}_{{xy}}-\beta (t){u}_{x}{u}_{y}=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}{v}_{t}-\alpha (t){v}_{{xx}}-\beta (t){\left({uv}\right)}_{x}=0,\end{eqnarray}$
where α(t) and β(t) are the real non-zero functions of t, u(x, y, t), and v(x, y, t) denote the real differentiable functions, while the subscripts denote the partial derivatives with respect to the scaled space variables x, y and time variable t [11]. Variable separation solutions for system (When α(t) = 1 and β(t) = 2, system (1 ) has been reduced to a (2+1)-dimensional modified dispersive water-wave system [12–14],2 ) has been used to describe the nonlinear dispersive long gravity waves propagating in two horizontal directions on shallow water with uniform depth [12–14].
$\begin{eqnarray}\begin{array}{l}{u}_{{ty}}+{u}_{{xxy}}-2{v}_{{xx}}-2{{uu}}_{{xy}}-2{u}_{x}{u}_{y}=0,\\ {v}_{t}-{v}_{{xx}}-2{\left({uv}\right)}_{x}=0,\end{array}\end{eqnarray}$
where u indicates the height for the water surface above the horizontal bottom and v indicates the horizontal velocity for the water wave. System (However, Painlevé analysis, auto-Bäcklund transformations and bilinear forms for system (1 ) have not been discussed. In section 2 , Painlevé analysis for system (1 ) will be investigated. In section 3 , auto-Bäcklund transformations and soliton solutions for system (1 ) will be constructed via the truncated Painlevé expansions. In section 4 , we will work out the bilinear forms for system (1 ). In section 5 , the interaction between the two solitons for system (1 ) will be analyzed via asymptotic analysis. Conclusions will be given in section 6 .
2. Painlevé analysis for system (1 )
Motivated by [15], solutions for system (1 ) can be expanded as the Laurent series:3 ) is assumed as4 ) into system (1 ) and balancing the highest-order nonlinear and linear terms, we work out1 ), we make the sum of the lowest power terms of φ in system (1 ) vanish. Due to the arbitrariness of uj and vj corresponding to the resonance point j, we get
$\begin{eqnarray}u=\displaystyle \sum _{j=0}^{+\infty }{u}_{j}{\phi }^{j+{\ell }},\,v=\displaystyle \sum _{j=0}^{+\infty }{v}_{j}{\phi }^{j+\unicode{x00237}},\end{eqnarray}$
where φ, uj's and vj's indicate the analytic functions with respect to x, y and t, j is a non-negative integer, while ℓ and ${\unicode{x00237}}$ denote the negative integers. The leading order of solutions ( $\begin{eqnarray}u\sim {u}_{0}{\phi }^{{\ell }},\,v\sim {v}_{0}{\phi }^{\unicode{x00237}},\end{eqnarray}$
where u0 and v0 denote the non-zero functions in the neighborhood of the non-characteristic movable singularity manifold. Substituting expressions ( $\begin{eqnarray}\begin{array}{l}{\ell }=-1,\,\unicode{x00237}=-2,\,{u}_{0}=\displaystyle \frac{2\alpha (t){\phi }_{x}}{\beta (t)},\\ {v}_{0}=-\displaystyle \frac{2\alpha (t){\phi }_{x}{\phi }_{y}}{\beta (t)}.\end{array}\end{eqnarray}$
Then, in order to find the resonance points, substituting $\begin{eqnarray}u\sim {u}_{0}{\phi }^{-1}+{u}_{j}{\phi }^{j-1},\,v\sim {v}_{0}{\phi }^{-2}+{v}_{j}{\phi }^{j-2},\end{eqnarray}$
into system ( $\begin{eqnarray}(j+1)(j-2){\left(j-3\right)}^{2}(j-4){\phi }_{x}^{4}{\phi }_{y}=0.\end{eqnarray}$
Thus, resonant points occur at j = −1, 2, 3, 3, 4.Setting8 ) into system (1 ), and making the coefficients of powers of φ vanish, we derive
$\begin{eqnarray}u=\displaystyle \sum _{j=0}^{4}{u}_{j}{\phi }^{j-1},\,v=\displaystyle \sum _{j=0}^{4}{v}_{j}{\phi }^{j-2},\end{eqnarray}$
substituting assumptions ( $\begin{eqnarray}\alpha ^{\prime} (t)\beta (t)-\beta ^{\prime} (t)\alpha (t)=0,\end{eqnarray}$
which is equivalent to $\begin{eqnarray}c=\displaystyle \frac{\alpha (t)}{\beta (t)},\end{eqnarray}$
where c denotes a real non-zero constant.Through the above calculation, we can obtain that one of u2(x, y, t) and v2(x, y, t), one of u4(x, y, t) and v4(x, y, t) as well as u3(x, y, t) and v3(x, y, t) are arbitrary. Hence, system (1 ) is Painlevé integrable under constraint (10 ).
3. Auto-Bäcklund transformations for system (1 )
Motivated by [15], we introduce the truncated Painlevé expansions for system (1 ) in the form of the Laurent series11 ) into system (1 ) and balancing the lowest-order nonlinear and linear terms, we obtain τ = 1 and η = 2. Substituting1 ), and making the coefficients of powers of φ vanish, we have1 ). With symbolic computation [6], we assume that $\varphi ={{\rm{e}}}^{{a}_{1}x+{a}_{2}y+{a}_{3}(t)}+1$ are in Bäcklund transformations (13 )–(17 ), and the solutions for system (1 ) are derived as:
$\begin{eqnarray}u={\varphi }^{-\tau }\displaystyle \sum _{{\imath }=0}^{\tau }{u}_{{\imath }}{\varphi }^{{\imath }},\,v={\varphi }^{-\eta }\displaystyle \sum _{\kappa =0}^{\eta }{v}_{\kappa }{\varphi }^{\kappa },\end{eqnarray}$
where τ and η indicate the positive integers, uı(x, y, t)'s, vκ(x, y, t)'s and φ(x, y, t) denote the analytic functions. Substituting expressions ( $\begin{eqnarray}u={\varphi }^{-1}\displaystyle \sum _{{\imath }=0}^{1}{u}_{{\imath }}{\varphi }^{{\imath }},\,v={\varphi }^{-2}\displaystyle \sum _{\kappa =0}^{2}{v}_{\kappa }{\varphi }^{\kappa },\end{eqnarray}$
into system ( $\begin{eqnarray}({\varphi }^{-4},{\varphi }^{-4}):\,{u}_{0}=\displaystyle \frac{2\alpha (t){\varphi }_{x}}{\beta (t)},{v}_{0}=-\displaystyle \frac{2\alpha (t){\varphi }_{x}{\varphi }_{y}}{\beta (t)},\,\end{eqnarray}$
$\begin{eqnarray}\,\begin{array}{lcl}({\varphi }^{-3},{\varphi }^{-3}) & : & 2\alpha (t){\varphi }_{x}{\varphi }_{{xy}}-\beta (t){u}_{1}{\varphi }_{x}{\varphi }_{y}\\ & & -\alpha (t){\varphi }_{y}{\varphi }_{{xx}}+{\varphi }_{y}{\varphi }_{t}-{v}_{1}\beta (t){\varphi }_{x}=0,\\ & & 2\alpha (t){\varphi }_{x}{\varphi }_{{xy}}+2\alpha (t){\varphi }_{{xx}}{\varphi }_{y}-\beta (t){v}_{1}{\varphi }_{x}\\ & & -2{\varphi }_{y}{\varphi }_{t}+2\beta (t){u}_{1}{\varphi }_{x}{\varphi }_{y}=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}({\varphi }^{-2},{\varphi }^{-2}) & : & \alpha (t){\varphi }_{{xxx}}{\varphi }_{y}-3\alpha (t){\varphi }_{{xxy}}{\varphi }_{x}\\ & & +{\varphi }_{{xx}}[\beta (t)({v}_{1}+{u}_{1}{\varphi }_{y})-\alpha (t){\varphi }_{{xy}}]-{\varphi }_{{xt}}{\varphi }_{y}\\ & & +{\varphi }_{{xy}}(2\beta (t){u}_{1}{\varphi }_{x}-{\varphi }_{t})\\ & & +{\varphi }_{x}[\beta (t)({\varphi }_{x}{u}_{1,y}+{\varphi }_{y}{u}_{1,x}+2{v}_{1,x})-{\varphi }_{{ty}}]=0,\\ & & 2{\alpha }^{2}(t){\varphi }_{{xxx}}{\varphi }_{y}+2{\alpha }^{2}(t){\varphi }_{{xxy}}{\varphi }_{x}\\ & & +2\alpha (t){\varphi }_{{xx}}\left[2\alpha (t){\varphi }_{{xy}}+\beta (t)\left({\varphi }_{y}{u}_{1}-\displaystyle \frac{{v}_{1}}{2}\right)\right]\\ & & +2\alpha (t)\beta (t){u}_{1}{\varphi }_{{xy}}{\varphi }_{x}-2\alpha (t){\varphi }_{{xt}}{\varphi }_{y}\\ & & -2\alpha (t){\varphi }_{{yt}}{\varphi }_{x}-\beta (t){v}_{1}{\varphi }_{t}\\ & & +\beta (t)\left\{{\varphi }_{x}[2\alpha (t){\varphi }_{y}{u}_{1,x}\right.\\ & & \left.+\beta (t){u}_{1}{v}_{1}]+2\alpha (t){v}_{2}{\varphi }_{x}^{2}\right\}=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}({\varphi }^{-1},{\varphi }^{-1}) & : & \beta (t){\varphi }_{x}{u}_{1,{xy}}+\beta (t){\varphi }_{{xx}}{u}_{1,y}+\beta (t){\varphi }_{{xy}}{u}_{1,x}\\ & & +\beta (t){u}_{1}{\varphi }_{{xxy}}-\alpha (t){\varphi }_{{xxxy}}\\ & & +\beta (t){v}_{1,{xx}}-{\varphi }_{{xyt}}=0,\\ & & {v}_{1,t}-2\alpha (t){v}_{2}{\varphi }_{{xx}}-2\alpha (t){\varphi }_{x}{v}_{2,{x}}\\ & & -\beta (t){v}_{1}{u}_{1,x}-\beta (t){u}_{1}{v}_{1,x}-\alpha (t){v}_{1,{xx}}=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}({\varphi }^{0},{\varphi }^{0}):{u}_{1,{ty}}+\alpha (t){u}_{1,{xxy}}-2\alpha (t){v}_{2,{xx}}-\beta (t){u}_{1}{u}_{1,{xy}}\\ \,-\beta (t){u}_{1,x}{u}_{1,y}=0,\\ \,{v}_{2,t}-\alpha (t){v}_{2,{xx}}-\beta (t){\left({u}_{1}{v}_{2}\right)}_{x}=0,\end{array}\end{eqnarray}$
where u1(x, y, t) and v2(x, y, t) can be regarded as the seed solutions for system ( $\begin{eqnarray}\begin{array}{l}u(x,y,t)={{ca}}_{1}\tanh \left[\displaystyle \frac{{a}_{1}x+{a}_{2}y+{a}_{3}(t)}{2}\right]+{{ca}}_{1},\\ v(x,y,t)=\frac{1}{2}{{ca}}_{1}{a}_{2}{{\rm{sech}} }^{2}\left[\displaystyle \frac{{a}_{1}x+{a}_{2}y+{a}_{3}(t)}{2}\right],\end{array}\end{eqnarray}$
where a1 and a2 are the real constants.4. Bilinear forms for system (1 )
Motivated by [16], we use the variable transformations19 ) into equation (1a ), under constraints (10 ), we get20 ) once with respect to x and y, respectively, and making the integration functions equal to 0, we work out
$\begin{eqnarray}\begin{array}{l}u=2\displaystyle \frac{\alpha (t)}{\beta (t)}{\left(\mathrm{ln}\displaystyle \frac{f}{g}\right)}_{x},\\ v=2\displaystyle \frac{\alpha (t)}{\beta (t)}{\left(\mathrm{ln}f\right)}_{{xy}}+\psi (y),\end{array}\end{eqnarray}$
where f(x, y, t) and g(x, y, t) denote the real differentiable functions of x, y and t, as well as ψ(y) is the real differentiable function of y. Substituting variable transformations ( $\begin{eqnarray}\begin{array}{l}2c{\left(\mathrm{ln}\displaystyle \frac{f}{g}\right)}_{{xyt}}+2c\alpha (t){\left(\mathrm{ln}\displaystyle \frac{f}{g}\right)}_{{xxxy}}\\ -4c\alpha (t){\left(\mathrm{ln}f\right)}_{{xxxy}}-4c\alpha (t){\left(\mathrm{ln}\displaystyle \frac{f}{g}\right)}_{x}{\left(\mathrm{ln}\displaystyle \frac{f}{g}\right)}_{{xxy}}\\ -4c\alpha (t){\left(\mathrm{ln}\displaystyle \frac{f}{g}\right)}_{{xx}}{\left(\mathrm{ln}\displaystyle \frac{f}{g}\right)}_{{xy}}=0.\end{array}\end{eqnarray}$
Integrating equation ( $\begin{eqnarray}\begin{array}{l}2c{\left(\mathrm{ln}\displaystyle \frac{f}{g}\right)}_{t}+2c\alpha (t){\left(\mathrm{ln}\displaystyle \frac{f}{g}\right)}_{{xx}}\\ -4c\alpha (t){\left(\mathrm{ln}f\right)}_{{xx}}-2c\alpha (t){\left[{\left(\mathrm{ln}\displaystyle \frac{f}{g}\right)}_{x}\right]}^{2}=0.\end{array}\end{eqnarray}$
According to the following formulas [16] $\begin{eqnarray}\begin{array}{l}{\left(\mathrm{ln}\displaystyle \frac{f}{g}\right)}_{x}=\displaystyle \frac{{D}_{x}f\cdot g}{{fg}},\,{\left(\mathrm{ln}\displaystyle \frac{f}{g}\right)}_{t}=\displaystyle \frac{{D}_{t}f\cdot g}{{fg}},\\ {\left(\mathrm{ln}f\right)}_{{xx}}=\displaystyle \frac{{D}_{x}^{2}f\cdot f}{2{f}^{2}},{\left(\mathrm{ln}f\right)}_{{xy}}=\displaystyle \frac{{D}_{x}{D}_{y}f\cdot f}{2{f}^{2}},\\ {\left(\mathrm{ln}f\right)}_{{yt}}=\displaystyle \frac{{D}_{y}{D}_{t}f\cdot f}{2{f}^{2}},\\ {\left(\mathrm{ln}\displaystyle \frac{f}{g}\right)}_{{xx}}=\displaystyle \frac{{D}_{x}^{2}f\cdot f}{2{f}^{2}}-\displaystyle \frac{{D}_{x}^{2}g\cdot g}{2{g}^{2}},\\ {\left(\mathrm{ln}\displaystyle \frac{f}{g}\right)}_{{xy}}=\displaystyle \frac{{D}_{x}{D}_{y}f\cdot f}{2{f}^{2}}-\displaystyle \frac{{D}_{x}{D}_{y}g\cdot g}{2{g}^{2}},\\ {\left(\mathrm{ln}\displaystyle \frac{f}{g}\right)}_{{yt}}=\displaystyle \frac{{D}_{y}{D}_{t}f\cdot f}{2{f}^{2}}-\displaystyle \frac{{D}_{y}{D}_{t}g\cdot g}{2{g}^{2}},\\ {\left(\mathrm{ln}\displaystyle \frac{f}{g}\right)}_{{xxy}}=\displaystyle \frac{{D}_{x}^{2}{D}_{y}f\cdot g}{{fg}}-\displaystyle \frac{{D}_{x}^{2}f\cdot g}{{fg}}\cdot \displaystyle \frac{{D}_{y}f\cdot g}{{fg}}\\ \,\,-2\displaystyle \frac{{D}_{x}{D}_{y}f\cdot g}{{fg}}\cdot \displaystyle \frac{{D}_{x}f\cdot g}{{fg}}\\ \,\,+2{\left(\displaystyle \frac{{D}_{x}f\cdot g}{{fg}}\right)}^{2}\cdot \displaystyle \frac{{D}_{y}f\cdot g}{{fg}},\end{array}\end{eqnarray}$
we obtain $\begin{eqnarray}[\alpha (t){D}_{x}^{2}-{D}_{t}]f\cdot g=0.\end{eqnarray}$
Dx, Dy and Dt are the Hirota bilinear operators defined as [16] $\begin{eqnarray}\begin{array}{l}{D}_{x}^{{l}_{1}}{D}_{y}^{{l}_{2}}{D}_{t}^{{l}_{3}}F(x,y,t)\cdot G(x,y,t)\\ \,=\,{\left(\displaystyle \frac{{\rm{\partial }}}{{\rm{\partial }}x}-\displaystyle \frac{{\rm{\partial }}}{{\rm{\partial }}{x{\rm{^{\prime} }}}^{}}\right)}^{{l}_{1}}{\left(\displaystyle \frac{{\rm{\partial }}}{{\rm{\partial }}y}-\displaystyle \frac{{\rm{\partial }}}{{\rm{\partial }}{y{\rm{^{\prime} }}}^{}}\right)}^{{l}_{2}}{\left(\displaystyle \frac{{\rm{\partial }}}{{\rm{\partial }}t}-\displaystyle \frac{{\rm{\partial }}}{{\rm{\partial }}{t{\rm{^{\prime} }}}^{}}\right)}^{{l}_{3}}\\ \times \,{\left.\left.F(x,y,t)G({x{\rm{^{\prime} }}}^{},{y{\rm{^{\prime} }}}^{},{t{\rm{^{\prime} }}}^{}\right)\right|}_{{x{\rm{^{\prime} }}}^{}=x,{y{\rm{^{\prime} }}}^{}=y,{t{\rm{^{\prime} }}}^{}=t},\end{array}\end{eqnarray}$
where F(x, y, t) is a function of x, y and t, $G(x^{\prime} ,y^{\prime} ,t^{\prime} )$ denotes a function of the independent variables of $x^{\prime} $, $y^{\prime} $ and $t^{\prime} $, while l1, l2 and l3 are the non-negative integers.Integrating equation (20 ) once with respect to x, and making the integration constant equal to 0, we work out19 ) into equation (1b ), under constraints (10 ), we have26 ) once with respect to x, and making the integration constant equal to 0, we obtain22 ) and equation (23 ) into equations (25 ) and (27 ), we get1 ) under constraint (10 ) as
$\begin{eqnarray}\begin{array}{l}2c{\left(\mathrm{ln}\displaystyle \frac{f}{g}\right)}_{{yt}}+2c\alpha (t){\left(\mathrm{ln}\displaystyle \frac{f}{g}\right)}_{{xxy}}-4c\alpha (t){\left(\mathrm{ln}f\right)}_{{xxy}}\\ \qquad \qquad -4c\alpha (t){\left(\mathrm{ln}\displaystyle \frac{f}{g}\right)}_{x}{\left(\mathrm{ln}\displaystyle \frac{f}{g}\right)}_{{xy}}=0.\end{array}\end{eqnarray}$
Substituting variable transformation ( $\begin{eqnarray}\begin{array}{l}2c{\left(\mathrm{ln}f\right)}_{{xyt}}-2c\alpha (t){\left(\mathrm{ln}f\right)}_{{xxxy}}\\ -\,4c\alpha (t){\left(\mathrm{ln}\displaystyle \frac{f}{g}\right)}_{{xx}}{\left(\mathrm{ln}f\right)}_{{xy}}-2\alpha (t)\psi (y){\left(\mathrm{ln}\displaystyle \frac{f}{g}\right)}_{{xx}}\\ -\,4c\alpha (t){\left(\mathrm{ln}\displaystyle \frac{f}{g}\right)}_{x}{\left(\mathrm{ln}f\right)}_{{xxy}}=0.\end{array}\end{eqnarray}$
Integrating equation ( $\begin{eqnarray}\begin{array}{l}2c{\left(\mathrm{ln}f\right)}_{{yt}}-2c\alpha (t){\left(\mathrm{ln}f\right)}_{{xxy}}-4c\alpha (t){\left(\mathrm{ln}\displaystyle \frac{f}{g}\right)}_{x}{\left(\mathrm{ln}f\right)}_{{xy}}\\ \,-2\alpha (t)\psi (y){\left(\mathrm{ln}\displaystyle \frac{f}{g}\right)}_{x}=0.\end{array}\end{eqnarray}$
Substituting formulae ( $\begin{eqnarray}\begin{array}{l}\left[2{{cD}}_{y}{D}_{t}-2\alpha (t){{cD}}_{x}^{2}{D}_{y}\right.\,\left.-\,4\alpha (t)\psi (y){D}_{x}\right]f\cdot g=0.\end{array}\end{eqnarray}$
Thus, we can derive the bilinear forms for system ( $\begin{eqnarray}\begin{array}{l}[\alpha (t){D}_{x}^{2}-{D}_{t}]f\cdot g=0,\\ [2{{cD}}_{y}{D}_{t}-2\alpha (t){{cD}}_{x}^{2}{D}_{y}-\,4\alpha (t)\psi (y){D}_{x}]f\cdot g=0.\end{array}\end{eqnarray}$
5. Soliton solutions for system (1 )
In order to derive certain N-soliton solutions for system (1 ), we give the following expansions:30 ) into bilinear forms (29 ) and making the coefficients zero on each power order of ϵ, we work out the N-soliton solutions for system (1 )
$\begin{eqnarray}\begin{array}{rcl}f & = & 1+\varepsilon {f}_{1}+{\varepsilon }^{2}{f}_{2}+{\varepsilon }^{3}{f}_{3}+\cdots +{\varepsilon }^{N}{f}_{N},\\ g & = & 1+\varepsilon {g}_{1}+{\varepsilon }^{2}{g}_{2}+{\varepsilon }^{3}{g}_{3}+\cdots +{\varepsilon }^{N}{g}_{N},\end{array}\end{eqnarray}$
where fϱ's and gϱ's (ϱ = 1, 2, 3, ⋯, N) denote the real functions of x, y and t, N is a positive integer, as well as ϵ is a real constant. Substituting expressions ( $\begin{eqnarray}\begin{array}{l}u=2c{\left(\mathrm{ln}\displaystyle \frac{f}{g}\right)}_{x},\\ f=\displaystyle \sum _{\mu =0,1}\exp \left\{\displaystyle \sum _{\iota =1}^{N}{\mu }_{\iota }[\mathrm{ln}({B}_{\iota })+{\eta }_{\iota }]+\displaystyle \sum _{\iota \lt \varsigma }^{(N)}{\mu }_{\iota }{\mu }_{\varsigma }\mathrm{ln}({A}_{\iota \varsigma })\right\},\\ v=2c{\left(\mathrm{ln}f\right)}_{{xy}}+\psi (y),\\ g=\displaystyle \sum _{\mu =0,1}\exp \left[\displaystyle \sum _{\iota =1}^{N}{\mu }_{\iota }{\eta }_{\iota }+\displaystyle \sum _{\iota \lt \varsigma }^{(N)}{\mu }_{\iota }{\mu }_{\varsigma }\mathrm{ln}({A}_{\iota \varsigma })\right],\end{array}\end{eqnarray}$
with $\begin{eqnarray*}\begin{array}{l}{\eta }_{\iota }={k}_{\iota }[x+{p}_{\iota }(y)+{\omega }_{\iota }(t)],\\ {p}_{\iota }(y)=\displaystyle \int \displaystyle \frac{{\left({B}_{\iota }-1\right)}^{2}\psi (y)}{2{B}_{\iota }{{ck}}_{\iota }^{2}}{\rm{d}}y,\\ {\omega }_{\iota }(t)=\displaystyle \int \displaystyle \frac{(1+{B}_{\iota })c\beta (t){k}_{\iota }}{({B}_{\iota }-1)}{\rm{d}}t,\\ {A}_{\iota \varsigma }=\displaystyle \frac{[({B}_{\iota }-1){k}_{\varsigma }-({B}_{\varsigma }-1){k}_{\iota }][{B}_{\iota }{k}_{\iota }({B}_{\varsigma }-1)-({B}_{\iota }-1){B}_{\varsigma }{k}_{\varsigma }]}{[{B}_{\iota }{k}_{\iota }({B}_{\varsigma }-1)-({B}_{\iota }-1){k}_{\varsigma }][({B}_{\iota }-1){B}_{\varsigma }{k}_{\varsigma }-({B}_{\varsigma }-1){k}_{\iota }]},\end{array}\end{eqnarray*}$
where Bι kι(Bς − 1) − (Bι − 1)kς ≠ 0, (Bι − 1)Bςkς − (Bς −1)kι ≠ 0, Bι ≠ 0, α(t) ≠ 0, kι ≠ 0, kι's and Bι's are the real constants, pι(y)'s denote the functions of y, ωι(t)'s are the functions of t, ∑μ=0,1 indicates a summation over all the possible combinations of μι = 0, 1 for ι = 1, 2,⋯, N, while ${\sum }_{\iota \lt \kappa }^{(N)}$ denotes a summation over all possible pairs (ι, κ) chosen from N elements under the condition ι < κ.When N = 2 in solutions (31 ), the second-order soliton solutions for system (1 ) can be expressed as
$\begin{eqnarray}u=2c{\left(\mathrm{ln}\displaystyle \frac{f}{g}\right)}_{x},\,v=2c{\left(\mathrm{ln}f\right)}_{{xy}}+\psi (y),\end{eqnarray}$
with $\begin{eqnarray*}\begin{array}{rcl}f & = & 1+{B}_{1}{{\rm{e}}}^{{\eta }_{1}}+{B}_{2}{{\rm{e}}}^{{\eta }_{2}}+{B}_{1}{B}_{2}{A}_{12}{{\rm{e}}}^{{\eta }_{1}+{\eta }_{2}},\,g=1+{{\rm{e}}}^{{\eta }_{1}}+{{\rm{e}}}^{{\eta }_{2}}+{A}_{12}{{\rm{e}}}^{{\eta }_{1}+{\eta }_{2}},\\ {A}_{12} & = & \displaystyle \frac{[({B}_{1}-1){k}_{2}-({B}_{2}-1){k}_{1}][{B}_{1}{k}_{1}({B}_{2}-1)-({B}_{1}-1){B}_{2}{k}_{2}]}{[{B}_{1}{k}_{1}({B}_{2}-1)-({B}_{1}-1){k}_{2}][({B}_{1}-1){B}_{2}{k}_{2}-({B}_{2}-1){k}_{1}]}.\end{array}\end{eqnarray*}$
Through asymptotic analysis of solutions (32 ), we analyze the interaction properties between the two solitons. We will introduce the symbols ${u}_{{\imath }}^{\pm }$'s and ${v}_{{\imath }}^{\pm }$'s (ı = 1, 2, 3), where the expressions before the interaction between the two solitons are denoted by ${u}_{{\imath }}^{-}$ and ${v}_{{\imath }}^{-}$, while the expressions after the interaction between the two solitons are expressed by ${u}_{{\imath }}^{+}$ and ${v}_{{\imath }}^{+}$.
Case 1: Elastic interaction
We assume that k2 > k1 > 0, $\tfrac{{\left({B}_{2}-1\right)}^{2}\psi (y)}{{B}_{2}{{ck}}_{2}^{2}}\gt \tfrac{{\left({B}_{1}-1\right)}^{2}\psi (y)}{{B}_{1}{{ck}}_{1}^{2}}$ and $\tfrac{({B}_{2}+1)\alpha (t){k}_{2}}{{B}_{2}-1}\gt \tfrac{({B}_{1}+1)\alpha (t){k}_{1}}{{B}_{1}-1}$, and then we work out the asymptotic expressions for solutions (32 )33 ) and (34 ), we can derive the velocity vectors of the ρth soliton as $\overrightarrow{{V}_{\rho }}={\left({V}_{\rho x},{V}_{\rho y}\right)}^{T}$, where33 ) and (34 ), we can get that the amplitudes of u1 and u2 are35 )–(37 ), we can work out that the velocity and amplitude for each soliton remain unchanged before and after the interaction, except for the phase shift associated with A12, as seen in figure 1.
$\begin{eqnarray}u\to \left\{\begin{array}{l}{u}_{1}^{-}=\displaystyle \frac{2{{ck}}_{1}({B}_{1}-1)}{1+{B}_{1}+2\sqrt{{B}_{1}}\cosh ({\eta }_{1}+\mathrm{ln}\sqrt{{B}_{1}})},\,({\eta }_{1}\sim 0,\,{\eta }_{2}\to -\infty ),\\ {u}_{1}^{+}=\displaystyle \frac{2{{ck}}_{1}({B}_{1}-1)}{1+{B}_{1}+2\sqrt{{B}_{1}}\cosh ({\eta }_{1}+\mathrm{ln}\sqrt{{A}_{12}{B}_{1}})},\,({\eta }_{1}\sim 0,\,{\eta }_{2}\to +\infty ),\\ {u}_{2}^{-}=\displaystyle \frac{2{{ck}}_{2}({B}_{2}-1)}{1+{B}_{2}+2\sqrt{{B}_{2}}\cosh ({\eta }_{2}+\mathrm{ln}\sqrt{{A}_{12}{B}_{2}})},\,({\eta }_{2}\sim 0,\,{\eta }_{1}\to +\infty ),\\ {u}_{2}^{+}=\displaystyle \frac{2{{ck}}_{2}({B}_{2}-1)}{1+{B}_{2}+2\sqrt{{B}_{2}}\cosh ({\eta }_{2}+\mathrm{ln}\sqrt{{B}_{2}})},\,({\eta }_{2}\sim 0,\,{\eta }_{1}\to -\infty ).\end{array}\right.\end{eqnarray}$
$\begin{eqnarray}v\to \left\{\begin{array}{l}{v}_{1}^{-}=\psi (y)\left[1+\displaystyle \frac{{\left({B}_{1}-1\right)}^{2}}{4{B}_{1}}{{\rm{sech}} }^{2}\left(\displaystyle \frac{{\eta }_{1}}{2}+\mathrm{ln}\sqrt{{B}_{1}}\right)\right],\,({\eta }_{1}\sim 0,\,{\eta }_{2}\to -\infty ),\\ {v}_{1}^{+}=\psi (y)\left[1+\displaystyle \frac{{\left({B}_{1}-1\right)}^{2}}{4{B}_{1}}{{\rm{sech}} }^{2}\left(\displaystyle \frac{{\eta }_{1}}{2}+\mathrm{ln}\sqrt{{A}_{12}{B}_{1}}\right)\right],\,({\eta }_{1}\sim 0,\,{\eta }_{2}\to +\infty ),\\ {v}_{2}^{-}=\psi (y)\left[1+\displaystyle \frac{{\left({B}_{2}-1\right)}^{2}}{4{B}_{2}}{{\rm{sech}} }^{2}\left(\displaystyle \frac{{\eta }_{2}}{2}+\mathrm{ln}\sqrt{{A}_{12}{B}_{2}}\right)\right],\,({\eta }_{2}\sim 0,\,{\eta }_{1}\to +\infty ),\\ {v}_{2}^{+}=\psi (y)\left[1+\displaystyle \frac{{\left({B}_{2}-1\right)}^{2}}{4{B}_{2}}{{\rm{sech}} }^{2}\left(\displaystyle \frac{{\eta }_{2}}{2}+\mathrm{ln}\sqrt{{B}_{2}}\right)\right],\,({\eta }_{2}\sim 0,\,{\eta }_{1}\to -\infty ).\end{array}\right.\end{eqnarray}$
From expressions ( $\begin{eqnarray}\begin{array}{l}{V}_{\rho x}=-\displaystyle \frac{4({B}_{\rho }+1){B}_{\rho }^{2}{c}^{2}{k}_{\rho }^{5}\beta (t)}{({B}_{\rho }-1)(4{B}_{\rho }^{2}{c}^{2}{k}_{\rho }^{4}+{\left({B}_{\rho }-1\right)}^{4}\psi {\left(y\right)}^{2})},\\ {V}_{\rho y}=-\displaystyle \frac{2({B}_{\rho }^{2}-1){B}_{\rho }{{ck}}_{\rho }^{3}\psi (y)\beta (t)}{4{B}_{\rho }^{2}{c}^{2}{k}_{\rho }^{4}+{\left({B}_{\rho }-1\right)}^{4}\psi {\left(y\right)}^{2}},\end{array}\end{eqnarray}$
where ρ is a positive integer. From expressions ( $\begin{eqnarray}\widetilde{{u}_{1}}=\displaystyle \frac{2{{ck}}_{1}(\sqrt{{B}_{1}}-1)}{\sqrt{{B}_{1}}+1},\,\widetilde{{u}_{2}}=\displaystyle \frac{2{{ck}}_{2}(\sqrt{{B}_{2}}-1)}{\sqrt{{B}_{2}}+1},\end{eqnarray}$
while we can obtain that the amplitudes of v1 and v2 are $\begin{eqnarray}\widetilde{{v}_{1}}=\displaystyle \frac{{\left({B}_{1}-1\right)}^{2}}{4{B}_{1}},\,\widetilde{{v}_{2}}=\displaystyle \frac{{\left({B}_{2}-1\right)}^{2}}{4{B}_{2}}.\end{eqnarray}$
From expressions (
Figure 1. Elastic interaction between two solitons via solutions ( |
Figure 2. Elastic interaction between two solitons via solutions ( |
Figure 3. Elastic interaction between two solitons via solutions ( |
Case 2: Inelastic interaction
In order to obtain the inelastic interaction between the two solitons, we take A12 equal to 0, that is, $\tfrac{{B}_{1}-1}{{k}_{1}}=\tfrac{{B}_{2}-1}{{k}_{2}}$ or $\tfrac{{B}_{1}-1}{{k}_{1}{B}_{1}}=\tfrac{{B}_{2}-1}{{k}_{2}{B}_{2}}$ in solutions (32 ). We take $\tfrac{{B}_{1}-1}{{k}_{1}}=\tfrac{{B}_{2}-1}{{k}_{2}}$, and then we derive the asymptotic expressions for solutions (32 )38 ), we can get that the amplitudes of ${u}_{1}^{-}$ and ${u}_{2}^{-}$ before the interaction are $\widetilde{{u}_{1}}=\tfrac{2{{ck}}_{1}(\sqrt{{B}_{1}}-1)}{\sqrt{{B}_{1}}+1}$ and $\widetilde{{u}_{2}}=\tfrac{2{{ck}}_{1}{\left(\sqrt{{B}_{2}}-1\right)}^{2}}{{B}_{1}-1}$, respectively, and the amplitude of ${u}_{3}^{+}$ after the interaction is $\widetilde{{u}_{3}}=\tfrac{2{{ck}}_{1}{\left(\sqrt{{B}_{1}}-\sqrt{{B}_{2}}\right)}^{2}}{{B}_{1}-1}$. $\widetilde{{u}_{1}}$, $\widetilde{{u}_{2}}$ and $\widetilde{{u}_{3}}$ have the following relationship:
$\begin{eqnarray}u\to \left\{\begin{array}{ll}{u}_{1}^{-}=\displaystyle \frac{2{{ck}}_{1}({B}_{1}-1)}{1+{B}_{1}+2\sqrt{{B}_{1}}\cosh ({\eta }_{1}+\mathrm{ln}\sqrt{{B}_{1}})}, & ({\eta }_{1}\sim 0,\,{\eta }_{2}\to -\infty ),\\ {u}_{2}^{-}=\displaystyle \frac{2{{ck}}_{1}{\left({B}_{2}-1\right)}^{2}}{({B}_{1}-1)\left(1+{B}_{2}+2\sqrt{{B}_{2}}\cosh ({\eta }_{2}+\mathrm{ln}\sqrt{{B}_{2}})\right)}, & ({\eta }_{2}\sim 0,\,{\eta }_{1}\to -\infty ),\\ {u}_{3}^{+}=\displaystyle \frac{2{{ck}}_{1}{\left({B}_{1}-{B}_{2}\right)}^{2}}{({B}_{1}-1)\left[{B}_{1}+{B}_{2}+2\sqrt{{B}_{1}{B}_{2}}\cosh \left({\eta }_{1}-{\eta }_{2}+\mathrm{ln}\sqrt{\tfrac{{B}_{1}}{{B}_{2}}}\right)\right]}, & \\ \quad ({\eta }_{1}-{\eta }_{2}\sim 0,\,{\eta }_{1}\to +\infty ,\,{\eta }_{2}\to +\infty ). & \end{array}\right.\end{eqnarray}$
$\begin{eqnarray}\begin{array}{c}v\to \left\{\begin{array}{l}{v}_{1}^{-}=\psi (y)\left[1+\displaystyle \frac{{\left({B}_{1}-1\right)}^{2}}{4{B}_{1}}{{\rm{{\rm{sech}} }}}^{2}\left(\frac{{\eta }_{1}}{2}+{\rm{ln}}\sqrt{{B}_{1}}\right)\right],\,({\eta }_{1}\sim 0,\,{\eta }_{2}\to -\infty ),\\ {v}_{2}^{-}=\psi (y)\left[1+\displaystyle \frac{{\left({B}_{2}-1\right)}^{2}}{4{B}_{2}}{{\rm{{\rm{sech}} }}}^{2}\left(\displaystyle \frac{{\eta }_{2}}{2}+{\rm{ln}}\sqrt{{B}_{2}}\right)\right],\,({\eta }_{2}\sim 0,\,{\eta }_{1}\to -\infty ),\\ {v}_{3}^{+}=\psi (y)\left[1+\displaystyle \frac{{\left({B}_{1}-{B}_{2}\right)}^{2}}{4{B}_{1}{B}_{2}}{{\rm{{\rm{sech}} }}}^{2}\left(\displaystyle \frac{{\eta }_{1}-{\eta }_{2}}{2}+{\rm{ln}}\sqrt{\displaystyle \frac{{B}_{1}}{{B}_{2}}}\right)\right],\\ ({\eta }_{1}-{\eta }_{2}\sim 0,\,{\eta }_{1}\to +\infty ,\,{\eta }_{2}\to +\infty ).\end{array}\right.\end{array}\end{eqnarray}$
From expression ( $\begin{eqnarray}\begin{array}{lcl}\widetilde{{u}_{3}} & = & \widetilde{{u}_{1}}+\widetilde{{u}_{2}}-2\sqrt{\widetilde{{u}_{1}}\widetilde{{u}_{2}}},\\ & & \mathrm{when}\,\displaystyle \frac{2{{ck}}_{1}(\sqrt{{B}_{1}}-1)(\sqrt{{B}_{2}}-1)}{{B}_{1}-1}\gt 0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{lcl}\widetilde{{u}_{3}} & = & \widetilde{{u}_{1}}+\widetilde{{u}_{2}}+2\sqrt{\widetilde{{u}_{1}}\widetilde{{u}_{2}}},\\ & & \mathrm{when}\,\displaystyle \frac{2{{ck}}_{1}(\sqrt{{B}_{1}}-1)(\sqrt{{B}_{2}}-1)}{{B}_{1}-1}\lt 0.\end{array}\end{eqnarray}$
In the case of $\tfrac{{B}_{1}-1}{{k}_{1}{B}_{1}}=\tfrac{{B}_{2}-1}{{k}_{2}{B}_{2}}$, we get the similar properties and asymptotic expressions.
In figure 4, the two solitons fuse into one, and the direction of propagation changes.
Figure 4. Inelastic interaction between two solitons via solutions ( |
Case 3: Soliton resonance
The resonance between the two solitons can be derived when A12 → +∞ in solutions (32 ), as seen in figure 5. From figure 5, we can observe that the two solitons initially fuse into one and then separate, and the amplitude of the interacting part is bigger than the amplitudes of the solitons in the other parts.
Figure 5. Resonance between two solitons via solutions ( |
When N = 3 in solutions (31 ), the third-order soliton solutions for system (1 ) can be expressed as
$\begin{eqnarray}u=2c{\left(\mathrm{ln}\displaystyle \frac{f}{g}\right)}_{x},\,v=2c{\left(\mathrm{ln}f\right)}_{{xy}}+\psi (y),\end{eqnarray}$
with $\begin{eqnarray*}\begin{array}{l}f=1+{B}_{1}{{\rm{e}}}^{{\eta }_{1}}+{B}_{2}{{\rm{e}}}^{{\eta }_{2}}+{B}_{3}{{\rm{e}}}^{{\eta }_{3}}\\ \,+\,{B}_{1}{B}_{2}{A}_{12}{{\rm{e}}}^{{\eta }_{1}+{\eta }_{2}}+{B}_{1}{B}_{3}{A}_{13}{{\rm{e}}}^{{\eta }_{1}+{\eta }_{3}}+{B}_{2}{B}_{3}{A}_{23}{{\rm{e}}}^{{\eta }_{2}+{\eta }_{3}}\\ \,+\,{B}_{1}{B}_{2}{B}_{3}{A}_{123}{{\rm{e}}}^{{\eta }_{1}+{\eta }_{2}+{\eta }_{3}},\\ g=1+{{\rm{e}}}^{{\eta }_{1}}+{{\rm{e}}}^{{\eta }_{2}}+{A}_{12}{{\rm{e}}}^{{\eta }_{1}+{\eta }_{2}}\\ \,+\,{A}_{13}{{\rm{e}}}^{{\eta }_{1}+{\eta }_{3}}+{A}_{23}{{\rm{e}}}^{{\eta }_{2}+{\eta }_{3}}+{A}_{123}{{\rm{e}}}^{{\eta }_{1}+{\eta }_{2}+{\eta }_{3}},\\ {A}_{123}={A}_{12}{A}_{13}{A}_{23}.\end{array}\end{eqnarray*}$
A study by [17] believed that the three-soliton condition implies the N-soliton condition.6. Conclusions
In this paper, a (2+1)-dimensional variable-coefficient modified dispersive water-wave system in fluid mechanics, i.e., system (1 ), has been investigated. It has been proved that system (1 ) is Painlevé integrable under constraint (10 ) via the Painlevé analysis. We have worked out auto-Bäcklund transformations (13 )–(17 ) and soliton solutions (18 ) via the truncated Painlevé expansions. Bilinear forms (29 ) and N-soliton solutions (31 ) have been derived. For the two solitons, interactions, inelastic interactions and soliton resonances via asymptotic analysis, as shown in figures 1, 4 and 5, respectively. From figures 1 and 2, we have determined that the velocities of the solitons are affected by the variable coefficient β(t). From figures 1 and 3, we have observed that the soliton shape and background plane are affected by ψ(y). In addition, those solitons could help people analytically study other nonlinear evolution equations in fluid mechanics, plasma physics, nonlinear dynamics, nonlinear optics and mathematical physics.