Some Interaction Solutions of a Reduced Generalised (3+1)-Dimensional Shallow Water Wave Equation for Lump Solutions and a Pair of Resonance Solitons

Abstract

Through Hirota bilinear transformation and symbolic computation with Maple, a class of lump solutions, rationally localised in all directions in the space, to a reduced generalised (3+1)-dimensional shallow water wave (SWW) equation are prensented. The resulting lump solutions all contain six parameters, two of which are free due to the translation invariance of the SWW equation and the other four of which must satisfy a nonzero determinant condition guaranteeing analyticity and rational localisation of the solutions. Then we derived the interaction solutions for lump solutions and one stripe soliton and the result shows that the particular lump solutions with specific values of the involved parameters will be drowned or swallowed by the stripe soliton. Furthermore, we extend this method to a more general combination of positive quadratic function and hyperbolic functions. Especially, it is interesting that a rogue wave is found to be aroused by the interaction between lump solutions and a pair of resonance stripe solitons. By choosing the values of the parameters, the dynamic properties of lump solutions, interaction solutions for lump solutions and one stripe soliton and interaction solutions for lump solutions and a pair of resonance solitons, are shown by dynamic graphs.

A more general form of lump solution, lumpoff, and instanton/rogue wave solutions of a reduced (3 + 1)-dimensional nonlinear evolution equation

“…The known lump solutions in Ref. [42], which contain nine parameters with three constraints and two non-zero conditions can be considered as a special case of our results.…”

A Predictable Rogue Wave and Generating Mechanisms

“…( 2). The known lump solutions in [35,36,38], which contain nine parameters with three constraints and two non-zero conditions, and [39,40] including two parameters can all be considered as special cases of our results.…”

Fusionable and fissionable waves of (2+1)-dimensional shallow water wave equation

“…The interaction solutions of nonlinear partial differential equations are a topic of general interest in nonlinear systems. [1][2][3][4] Among them, shallow water wave equation has been one of the hottest issues in recent years, [5][6][7][8][9][10][11] such as marine engineering, hydrodynamics, mathematical physics in other fields. Because its exact solution is a special solution existing stably in space, [12] it has very important practical significance for many complex physical phenomena [13] and some nonlinear engineering problems.…”

“…The main purpose of this article is to study the fusion and fission waves and some interaction solutions of the (2 + 1)dimensional SWW equation, [5] which is usually written as u yt − u xxxy − 3u xx u y − 3u x u xy + u xx = 0.…”