Application of the Bilinear Neural Network Approach to Kadomtsev-Petviashvili Model for Nonlinear Waves

Abstract

Nonlinear evolution equations (NLEEs) serve as fundamental models for wave phe-nomena in many physical systems, including fluid dynamics, plasma physics, optical pulse propagation, and atmospheric processes. They describe the spatiotemporal evolu-tion of waves in media where nonlinear interactions and dispersive effects act together. A prominent feature of such equations is the emergence of solitons, which are localized travelling waves capable of preserving their shape and speed over long distances. This remarkable property is generally attributed to a balance between nonlinear steepening and dispersive spreading. Solitons appear across a broad spectrum of physical contexts, ranging from hydrodynamics, nonlinear optics, and plasma physics to solid-state and lat-tice systems, where they often display particle-like behavior while retaining their wave nature. They have also been observed in more complex settings such as astrophysical, condensed-matter, nuclear, and other nonlinear media, highlighting their wide applica-bility and importance in both theoretical studies and practical applications of nonlinear wave dynamics. Different types of solitons exist based on their shape and behavior. Kink solitons rep-resent a transition from one state to another, similar to steps in a staircase. Breather solitons are localized waves that expand and contract over time. Rogue waves are ex-tremely large and sudden waves that can be much higher than normal waves. Lump solitons are localized in all directions and gradually disappear with distance. These d-ifferent wave structures help researchers better understand nonlinear wave behavior in real-world systems. This thesis presents a bilinear neural network modeling (BNNM) framework that cou-ples the Hirota bilinear approach with trainable neural parameters to obtain closed-form solutions of the (3+1)-dimensional generalized Kadomtsev-Petviashvili equation