Chaotic and solitonic solutions for a new time-fractional two-mode korteweg-de vries equation

The two-mode Korteweg-de Vries (TMKdV) equation is a nonlin-ear dispersive wave model that describes the motion of two different directional wave modes with the same dispersion relations but with various phase velocities, nonlinearity, and dispersion parameters. In this work, we study the dynamics of the model analytically in a time-fractional sense to ensure the stability of the extracted waves of the TMKdV equation. We use the fractional power series technique to conduct our analysis. We show that there is a homotopy mapping of the solution as the Caputo time-fractional derivative order varies over (0, 1] and that both waves have the same physical shapes but with reflexive relation.