‎‎ ‎ Analytical and Computational Study of Solitary Wave Solutions Using the Kudryashev Method with Neural Parameter Optimization

Abstract

Nonlinear partial differential equations (PDEs) play a fundamental role in modeling various complex physical phenomena such as fluid dynamics, plasma physics, and nonlinear optics. Among these, the  equation and its generalizations have attracted considerable attention due to their rich soliton dynamics. In this study, we employ the Kudryashev method to derive analytical solitary wave solutions of Burgers–Korteweg–de Vries (BKdV) equation and combine it with neural-based optimization to estimate unknown model parameters accurately. The Kudryashev approach, which expresses nonlinear solutions through rational functions of exponential forms, allows an efficient representation of the soliton profiles. The proposed integration of the Kudryashev analytical framework with data-driven parameter learning not only validates the classical soliton structures but also enhances solution precision. This hybrid methodology offers a promising tool for solving a broad class of nonlinear PDEs in applied mathematics and mathematical physics, bridging the gap between symbolic and computational techniques.