Wave Propagation and Chaotic Behavior in Conservative and Dissipative Sawada–Kotera Models

This paper presents both analytical and numerical studies of the conservative Sawada–Kotera equation and its dissipative generalization, equations known for their soliton solutions and rich chaotic dynamics. These models offer valuable insights into nonlinear wave propagation, with applications in fluid dynamics and materials science, including systems such as liquid crystals and ferrofluids. It is shown that the conservative Sawada–Kotera equation supports traveling wave solutions corresponding to elliptic limit cycles, as well as two- and three-dimensional invariant tori surrounding these cycles in the associated ordinary differential equation (ODE) system. For the dissipative generalized Sawada–Kotera equation, chaotic wave behavior is observed. The transition to chaos in the corresponding ODE system follows a universal bifurcation scenario consistent with the framework established by FShM (Feigenbaum-Sharkovsky-Magnitskii) theory. Notably, this study demonstrates for the first time that the conservative Sawada–Kotera equation can exhibit complex quasi-periodic wave solutions, while its dissipative counterpart admits an infinite number of stable periodic and chaotic waveforms.