<abstract> <p>In this study, a fourth-order nonlinear wave equation with variable coefficients was investigated. Through appropriate choice of the free parameters and using the simplified linear superposition principle (LSP) and velocity resonance (VR), the examined equation can be considered as Hirota–Satsuma–Ito, Calogero–Bogoyavlenskii–Schiff and Jimbo–Miwa equations. The main objective of this study was to obtain novel resonant multi-soliton solutions and investigate inelastic interactions of traveling waves for the above-mentioned equation. Novel resonant multi-soliton solutions along with their essential conditions were obtained by using simplified LSP, and the conditions guaranteed the existence of resonant solitons. Furthermore, the obtained solutions were used to investigate the dynamic and fission behavior of Y-type multi-soliton waves. For an accurate investigation of physical phenomena, appropriate free parameters were chosen to ascertain the impact on the speed of traveling waves and the initiation time of fission. Three-dimensional and contour plots of the obtained solutions are presented in <xref ref-type="fig" rid="Figure1">Figures 1</xref>–<xref ref-type="fig" rid="Figure6">6</xref>. Additionally, two nonlinear equations were formulated and investigated using VR, and the related soliton molecules were simultaneously extracted. The reported resonant Y-type multi-soliton waves and equations are new and have not been previously investigated. They can be used to explain modeled physical phenomena and can provide information about dynamic behavior of shallow water waves.</p> </abstract>

It is renowned that Hirota–Satsuma–Ito (HSI) equation is widely used to study wave dynamics of shallow water. In this work, two new HSI-like equations are investigated which could be extended to diversify problems in natural phenomena and give admirable contributions by applying the generalized exponential rational function method (GERFM). With the aid of symbolic calculations, various constraints on the free parameters are given, while classes of wave solutions are explicitly constructed from the coefficients of the combined non-linear and dissipative terms. After specifying values for free parameters, singular, periodic singular and anti-kink waves are demonstrated in 3D figures to exhibit different kinds of wave propagations. The fact that parameters directly influence the wave amplitude and speed of traveling waves is illustrated. The derived results are innovative and have important applications in the current field of mathematical physics research. Eventually, we show that generalized exponential rational function method is effective and straightforward to solve higher-order and high-dimensional non-linear evolution equations.