Traveling Wave Solutions for an Initial Value Problem of Generalized Fractional Nonlinear Korteweg–de Vries Equations

ABSTRACT This paper comprehensively investigates the existence, uniqueness, and stability of traveling wave solutions for an initial value problem of generalized nonlinear Korteweg–de Vries equations of fractional order. We apply the Banach contraction principle and Schauder's fixed‐point theorem to establish the existence and uniqueness of solutions. Furthermore, we examine the stability of the solutions using Ulam–Hyers theorems. Two well‐detailed examples and one explicit solution are provided to illustrate the practical applicability and validity of our theoretical results. The solution's parameter conditions detail the amplitude, energy, wavelength, frequency, and propagation characteristics of waves. These parameters capture the intricate balance between nonlinear and dispersive effects that shape the wave phenomena modeled by the Korteweg–de Vries equation. The solution indicates that as the amplitude parameter increases, wave height also rises due to a higher wave number (shorter wavelength) and nonlinear amplification, resulting in more waves within a given interval. The amplitude further increases with the wave number due to higher energy levels associated with shorter wavelengths.