The nonlinear vibration and dispersive wave systems with extended homoclinic breather wave solutions

Abstract

This article investigates the extended homoclinic (heteroclinic) breather wave solutions and interaction periodic and dark soliton solutions to the nonlinear vibration and dispersive wave systems. The solutions include periodic, breather, and soliton solutions. The bilinear form is considered in terms of Hirota derivatives. Accordingly, we utilize the Cole–Hopf algorithm to obtain the exact solutions of the ( 2 + 1 2+1 )-dimensional modified dispersive water-wave system. The analytical treatment of extended homoclinic breather wave solutions and interaction periodic and dark soliton solutions are studied and plotted in four forms of density plots. A nonlinear vibration system will be studied. Employing appropriate mathematical assumptions, the novel kinds of the extended homoclinic breather wave solutions and interaction periodic and dark soliton solutions are derived and constructed in view of the combination of kink, periodic, and soliton for an extended homoclinic breather and also a combination of two kinks, periodic and dark soliton in terms of exponential, trigonometric, hyperbolic functions for interaction periodic and dark soliton of the governing equation. To achieve this, the illustrative example of the ( 2 + 1 2+1 )-D modified dispersive water-wave system is furnished to demonstrate the feasibility and reliability of the procedure applied in this research. The trajectory solutions of the traveling waves are offered explicitly and graphically. The effect of the free parameters on the behavior of designed figures of a few obtained solutions for two nonlinear rational exact cases was also considered. By comparing the suggested scheme with the other existing methods, the results state that the execution of this technique is succinct, extensive, and straightforward.

Conserved vectors and symmetry solutions of the Landau–Ginzburg–Higgs equation of theoretical physics

“…Nonlinear partial differential equations (NLPDEs) are extensively employed in the fields of science and engineering to represent and study a wide range of nonlinear phenomena encountered in practical applications [1][2][3][4][5][6][7][8][9][10][11][12][13]. These equations play a crucial role in modeling various scenarios, including fluid dynamics problems, wave propagation in complex media, seismic wave analysis, and the characterization of optical fibers.…”

“…They focused on obtaining double-periodic soliton solutions. In a separate study, Rao et al [9] utilized the Cole-Hopf algorithm to analyze the (2+1)-dimensional modified dispersive water-wave system, and obtained its exact solutions. Zhang and Wang [10] employed various methods, including the simplified extended tanh-function method, variational method, and He's frequency formulation method, to investigate the Fokas system that arises in monomode optical fibers.…”

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].…”