A Mathematical Study of the (3+1)-D Variable Coefficients Generalized Shallow Water Wave Equation with Its Application in the Interaction between the Lump and Soliton Solutions

Abstract

In this paper, the Hirota bilinear method, which is an important scheme, is used. The equation of the shallow water wave in oceanography and atmospheric science is extended to (3+1) dimensions, which is a well-known equation. A lot of classes of rational solutions by selecting the interaction between a lump and one- or two-soliton solutions are obtained. The bilinear form is considered in terms of Hirota derivatives. Accordingly, the logarithm algorithm to obtain the exact solutions of a (3+1)-dimensional variable-coefficient (VC) generalized shallow water wave equation is utilized. The analytical treatment of extended homoclinic breather wave solutions is studied and plotted in three forms 3D, 2D, and density plots. Using suitable mathematical assumptions, the established solutions are included in view of a combination of two periodic and two solitons in terms of two trigonometric and two hyperbolic functions for the governing equation. Maple software for computing the complicated calculations of nonlinear algebra equations is used. The effect of the free parameters on the behavior of acquired figures to a few obtained solutions for two nonlinear rational exact cases was also discussed.

Exact soliton solutions and the significance of time-dependent coefficients in the Boussinesq equation: theory and application in mathematical physics

“…Also researchers have developed several effective, powerful and efficient exact methods for uncovering the solutions of these equations. Notable techniques encompass the Jacobi elliptic function expansion scheme 30 , the Hirota bilinear method 31 , the exp-function method 32 , the new extended algebraic method 33 , the unified method 34 , the -expansion technique 35 , the auxiliary equation outline 36 , the Darboux transformation technique 37 , the Bäcklund transformation 38 , the modified extended tanh technique with Riccati equation 39 , the generalized -expansion approach 40 , the modified Kudryashov method 41 – 43 , the sine–Gordon expansion method 44 , the modified sine–cosine method 45 , the consistent Riccati expansion solvability technique 46 , the modified sine–Gordon expansion approach 47 , the modified simple equation method 48 – 50 , the generalized Kudryashov method 51 – 53 , among others.…”

A numerical aproach to dispersion-dissipation-reaction model: third order KdV-Burger-Fisher equation

“…Researchers can plan and conduct experiments by creating the suitable environmental conditions in order to find these parameters or functions. In the recent past, a variety of techniques have been proposed suh as the improved mother optimization algorithm [51], a promoted Remora optimization algorithm [52], a dwarf mongoose optimization algorithm [53], multi-criteria evaluation and optimization [54], training-based optimization algorithm [55], an improved water wave optimization algorithm [56], amended Dragon Fly optimization algorithm [57], vulture optimization algorithm [58], fractional order version of dragonfly algorithm [59], a new biomass-based hybrid energy system [60], ranking extreme efficient decision [61], k-lump and k-kink solutions [62], shallow water wave equation [63], N-lump and interaction solutions [64], lump and their interactions [65][66][67] and other interesting subjects in nonlinear sciences and engineering [68][69][70][71].…”