The bilinear neural network method for solving Benney–Luke equation

Benney–Luke equation, the estimation of water wave propagation on the water’s surface, is significantly important in studying the tension of water waves in physics. This paper focuses on the Bilinear Neural Network method (BNNM) to find the solutions to the Benney–Luke equation. Using the Hirota bilinear operator, the Benney–Luke equation is transformed into a bilinear expression and establishes models of neurons. The method construction is straightforward, well-organized, and effective for finding various exact analytic solutions of the Benny–Luke equation consisting of one-soliton solutions, two-soliton solutions, one-soliton hyperbolic wave solutions, two-soliton hyperbolic wave solutions, mixed-soliton hyperbolic solutions, and mixed-multi hyperbolic solutions by choosing the appropriate test functions. The solutions are rigorously depicted in three dimensions to observe the mutable behavior.