Investigation of multiple soliton solutions for high-dimensional models using automatic logarithmic bilinear transformation

Abstract

The high-dimensional Fokas model can describe a wide range of complex nonlinear wave phenomena; however, traditional analytical methods struggle to obtain a diverse set of exact analytical solutions, which limits the theoretical analysis and engineering applications of such models. In this paper, we address the problem of finding exact solutions for the high-dimensional Fokas model by combining Maple’s automatic bilinearization technique with an exploratory method using bilinear neural networks. Leveraging the high precision of symbolic computation and the powerful fitting capabilities of neural networks, we obtained a variety of analytical solutions for this model, including soliton solutions, soliton interaction solutions, soliton interaction solutions in a periodic background, periodically modulated soliton solutions, and traveling wave solutions. Compared to previous studies, this work marks the first application of Maple’s automatic bilinearization technique to this high-dimensional model, thereby enriching the known set of solutions for this model. After applying a characteristic scale transformation to the obtained dimensionless analytical solutions, physically meaningful solutions can be recovered. The relevant results provide a theoretical reference for the analysis of nonlinear wave theory in the fields of fluid mechanics and ocean engineering.