Uncovering the soliton solutions and interaction aspects of combined CBS-nCBS model utilizing Bäcklund transform

Abstract

The Calogero-Bogoyavlenskii-Schiff (CBS) equation is a nonlinear integrable partial differential equation that plays a significant role in studying solitons, wave interactions, and various physical phenomena, including fluid dynamics and optical fibers. This work focuses on solving the (3+1)-dimensional combined CBS-negative-order CBS (CBS-nCBS) equation using multiple analytical techniques. First, the Hirota bilinear form is employed to derive the Bäcklund transformation through different exchange identities. This transformation yields exponential and rational function solutions, enabling the identification of singular and kink solitons. Additionally, by applying various ansatzes within the bilinear framework, we construct two-wave, three-wave, multi-wave, and breather solutions, providing deeper insights into the equation’s characteristics. Furthermore, the given model is transformed into an ordinary differential equation via a traveling wave transformation. The \(G'/(bG' + G + a) \) method is then applied to extract solutions involving trigonometric and hyperbolic functions, leading to periodic and kink solitons. The graphical representations presented in this study are instrumental in visualizing the intricate dynamics of these solutions. The findings enhance our understanding of the diverse soliton structures and broaden the potential applications of the CBS equation.