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To analyze and study the behaviour of the shallow water waves, the perturbed Boussinesq equation has acquired fundamental importance. The principal objective of this paper is to manifest the exact traveling wave solution of the perturbed Boussinesq equation by two well known techniques named as, two variables (G′G,1G) expansion method and generalized projective Riccati equations method. A diverse array of soliton solutions, encompassing periodic, bright solitons, singular solitons and bright singular solitons are obtained by the applications of proposed techniques. The constraint conditions for newly constructed solutions are also specified. To enhance comprehension, the numerical illustrations of constructed solutions have been represented using surface plots, 2D plots and density plots. The results delineated in this paper transcend existing analysis, offering a novel, well-structured, and modern perspective. The solutions obtained not only enrich understanding of shallow water wave models but also exhibit efficacy in providing detailed descriptions of their dynamics. [ABSTRACT FROM AUTHOR]

In this article, the generalized (3+1) dimensional Kadomtsev–Petviashvili (KP) and modified Kadomtsev–Petviashvili equations are explored, along with weak non-linearity, dispersion and disturbances which can demonstrate the expansion of surface water and prolonged waves in fluid dynamics. These models explain numerous nonlinear phenomena in the field of fluid dynamics, plasma physics and many more. Modified auxiliary equation method is implemented to derive analytic exact solutions for the governing equations. Some interesting and new travelling wave patterns have been observed. The obtained results include kink soliton, kinky periodic solitary wave, dark-bright soliton and periodic waves. Furthermore, graphical analysis is performed by selecting appropriate values of parameters in these solutions to explain the dynamic behavior of some different types of solitons. The proposed technique is well organized and proficient to discuss various KP-type equations physically. [ABSTRACT FROM AUTHOR]