Exact soliton, lump, and breather solutions of the (3 + 1)-dimensional Jimbo-Miwa equation via the bilinear neural network method

Abstract

Abstract The Jimbo-Miwa (3 + 1)-dimensional nonlinear equation is a well-known high-dimensional extension of integrable models and is widely used to describe nonlinear wave behavior in plasma physics, fluid dynamics, nonlinear optics, and quantum field theory. In this paper, we present a structured symbolic approach based on the Bilinear Neural Network Method (BNNM) to address the complexities associated with solving this equation. The proposed framework combines two complementary approaches: deriving exact soliton solutions through the Hirota bilinear method and developing a neural network-based scheme to obtain analytical approximations of the Jimbo-Miwa dynamics. Within this framework, we introduce several neural network ansatz structures, including 4-2-1, 4-3-1, and deep 4-2-2-1 architectures, which allow for the systematic construction of broad families of exact analytical solutions. Our results include explicit rational lump solutions, periodic breather solutions, and rich hybrid interactions between lumps and solitons. This combined approach not only recovers known solution forms but also provides greater flexibility for exploring solution spaces that are difficult to handle with conventional methods. Furthermore, the neural network framework offers an effective route to approximate analytical solutions of the Jimbo–Miwa model, improving computational efficiency and enhancing predictive capability in application-oriented settings.