Abstract
A generalized (2+1)-dimensional variable-coefficient KdV equation is introduced, which can describe the interaction between a water wave and gravity-capillary waves better than the (1+1)-dimensional KdV equation. TheN-soliton solutions of the (2+1)-dimensional variable-coefficient fifth-order KdV equation are obtained via the Bell-polynomial method. Then the soliton fusion, fission, and the pursuing collision are analyzed depending on the influence of the coefficienteAij; wheneAij=0, the soliton fusion and fission will happen; wheneAij≠0, the pursuing collision will occur. Moreover, the Bäcklund transformation of the equation is gotten according to the binary Bell-polynomial and the period wave solutions are given by applying the Riemann theta function method.
Highlights
In soliton theory, the nonlinear evolution equations (NLEEs) [1,2,3] have described natural phenomena in many aspects, such as in the nonlinear phonology [4, 5], water waves [6], hydromechanics [7], and super symmetry
In the hydromechanics, the nonlinear evolution equations can explain the interaction of the waves by the different dispersion relations
As for the elastic, the amplitudes, velocities, and shapes of the soliton can be brought into correspondence with the initial soliton, but for the inelastic collision, after the interaction, one soliton can be divided into two or more solitons, a phenomenon called soliton fission, or contrarily, two or more solitons can be merged into one soliton which is called soliton fusion
Summary
The nonlinear evolution equations (NLEEs) [1,2,3] have described natural phenomena in many aspects, such as in the nonlinear phonology [4, 5], water waves [6], hydromechanics [7], and super symmetry. We introduced a generalized variablecoefficient fifth-order KdV equation as follows: ut + a (t) uux + b (t) uxxx + c (t) u2ux + d (t) uxu2x. The focus of the paper is to get the N-soliton solutions of the generalized variable-coefficient fifth-order (2 + 1)dimensional equation and analyze the interaction of the water wave and the gravity-capillary wave [26,27,28,29,30]. The details of the paper are as follows: Section 2 introduces a variablecoefficient fifth-order (2 + 1)-dimensional KdV equation. We explain the soliton fission and fusion and the soliton pursuing collision of the variable-coefficient fifth-order (2 + 1)-dimensional KdV equation according to the different coefficients eAij. the Backlund transformation is given
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