Bilinear neural network method for solving extended (2+1)-dimensional sixth-order benney-luke equation

Abstract

The Benney–Luke (BL) equation is a fundamental nonlinear evolution equation that models long-wave propagation in fluid dynamics and other nonlinear dispersive media. To understand more complex wave interactions and higher-order dispersive effects, extended forms of the BL equation have been developed, providing improved physical realism in describing nonlinear wave phenomena. Despite these advancements, existing studies on the extended sixth-order BL equation remain limited in the systematic construction of exact analytical solutions, particularly those encompassing diverse nonlinear structures such as rogue waves, lump waves, and peak-type solutions. This paper proposes a novel roadmap of the bilinear neural network method to derive new classes of exact solutions for the extended sixth-order Benney–Luke (BL) equation. Extended from previous models due to space-lower nonlinearities, the extended sixth-order equation incorporates additional dispersive and nonlinear interaction terms, enabling more comprehensive modeling of wave dynamics in fluid systems and nonlinear phenomena. Using Hirota’s bilinear operator and a neural network-based framework, we construct a diverse spectrum of analytical solutions, including kink, rogue, lump, and peakon-type solitons. These solutions significantly expand the known solution space of the BL family and offer deeper physical insights into nonlinear wave behavior such as wave steepening, resonance, and dispersion. The extended equation not only bridges mathematical rigor and physical realism but also improves the computational efficiency and adaptability of neural network-based structures in analyzing nonlinear partial differential equations.

Keywords

Sixth-order Benney-Luke equation bilinear neural network method / Hirota bilinear operator / (2+1)-dimensional Benney-Luke equation / Soliton solution / Analytic exact solution

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