Analyzing soliton solutions and multi-structured solitons in the KdV-SKR Equation in fluids using three novel approaches

In this work, we investigate a variety of soliton solutions for the (2+1)-dimensional Korteweg-de Vries-Sawada-Kotera-Ramani (KdV-SKR) equation, which is a significant class of nonlinear evolution equations that arise in fluid dynamics and plasma physics. This equation is a combination of the SK and KdV equations. Obtaining exact solutions to this equation presents a significant analytical challenge due to its multidimensional nature and high degree of nonlinearity. However, three newly advanced methodologies, viz., the Hirota bilinear (HB) method, the generalized exponential differential rational function (GEDRF) approach, and the improved modified Sardar sub-equation (IMSSE) approach, can be employed to tackle this challenge. We employ a logarithmic transformation to obtain the bilinear equation in terms of an auxiliary function and subsequently formulate it in HB form to derive various soliton solutions. Furthermore, we derive new families of distinct soliton solutions expressed in trigonometric, hyperbolic, rational, and exponential functions using the GEDRF and IMSSE techniques. The obtained solutions provide graphical representations and physical relevance of the wave profiles, including lump, breather, multi-wave, periodic-lump interactions, solitary-lump interactions, periodic singular, bright, dark, and singular soliton solutions. These solitonic waves show the dynamic properties of the derived solutions displayed through 2D, 3D, density, and contour graphics using Maple software. Solitons and solitonic structures are localized waves that occur across a wide range of disciplines, including plasma physics, oceanography, nonlinear sciences, atmospheric sciences, and earth sciences.