Solitary wave solutions for the (1+1)-dimensional nonlinear Kakutani–Matsuuchi model of internal gravity waves

Abstract

The Kakutani–Matsuuchi (KM) equation provides a mathematical framework for studying the dynamics of extended nonlinear internal gravity waves in a stratified fluid medium, including important phenomena such as solitary wave propagation. This study explores various analytical and semi-analytical methods to obtain accurate solutions for the (1＋1)-dimensional KM model.The primary objective of this research is to derive new solitary wave and periodic wave solutions, and subsequently validate these analytical techniques by comparing them with solutions obtained through the Adomian decomposition method. The KM model governs the dynamics of nonlinear waves and exhibits significant connections with other nonlinear equations, notably the (KdV) equation. The analytical methods, particularly the Khater II and auxiliary equation sub-equation methods, lead to closed-form solutions by reducing the partial differential equation to a system of ordinary differential equations. Conversely, the Adomian method provides a semi-analytical solution expressed as an infinite series.Solitary and periodic traveling wave solutions are systematically derived using the aforementioned analytical techniques. To verify their accuracy, solutions obtained through Adomian decomposition are computed recursively and compared. The results highlight the effectiveness of the analytical methods in discovering new exact solutions, thereby enhancing our understanding of the physical phenomena described by the KM equation. These solutions contribute to a more nuanced understanding of the properties and behaviors of internal waves, thereby enriching our overall knowledge in this field. Consequently, this research further develops analytical solution techniques tailored for nonlinear wave models prevalent in fluid dynamics.