Characteristics of lump-kink and their fission-fusion interactions, rogue, and breather wave solutions for a (3+1)-dimensional generalized shallow water equation
Dipankar Kumar, Irfan Raju, Gour Chandra Paul, Md. Emran Ali, Md. Dalim Haque
Abstract
In this study, lump and two classes of interaction, multi-stripe, and breather wave solutions for the (3+1)-dimensional generalized shallow water equation are presented via the Hirota bilinear method. Interaction solutions are found between one-lump and one-stripe, and one-lump and two-stripes solutions by combining a quadratic function and an exponential function, and a quadratic function and a hyperbolic cosine or double exponential functions, respectively. Dynamical behaviours of some obtained valid solutions are presented through some graphs. The physical interpretation of fission-fusion dynamics is also explained graphically through lump-kink interaction solutions. During the fission-fusion interaction process, it is seen that stripe solitons split into a stripe and a lump soliton, and then the lump and stripe solitons fuse together. During this process, a rogue wave is found between one lump and twin stripes soliton at t=0. Furthermore, multi-stripe and breather wave solutions are investigated by choosing the appropriate functions and the values for the free parameters. The multi-stripe waves are found to be nonsingular and rectangular hyperbolic shaped. On the other hand, breather waves are found to be periodic, which can evolve periodically along a straight line in the xy-plane. The produced wave solutions might be helpful to understand the propagation behaviour of waves in shallow water.