On Exploration of Soliton Solutions for the Higher Dimensional Dissipative Zabolotskaya–Khokhlov Equation by Using Modified Extended Direct Algebraic Neural Networking

This work develops a new computational algorithm. The modified extended direct algebraic neural network method (MEDANNM) is a synergistic combination of algebraic geometry and deep learning to find soliton solutions. The higher-dimensional dissipative Zabolotskaya–Khokhlov equation is a paradigm model for studying nonlinear wave propagation. This nonlinear model is very crucial in shock formation and energy dissipation in multidimensional acoustic and optical media. The MEDANNM uses a direct algebraic parameterization of the soliton profiles, which converge rapidly and with high accuracy. The implementation will focus on a double-hidden-layer [4-2-2-1] neural network model. The input layer accepts spatiotemporal variables and dissipation parameters; two intermediate hidden layers apply algebraic constraints; and the output layer provides the soliton amplitude. The activation in the first hidden layer is uniquely based on the solution manifold of the modified extended direct algebraic method, thereby incorporating algebraic accuracy into the learning mechanisms and improving the fidelity of the solution and revealing new soliton structures, such as traveling, breather-like, and bright-dark. Visual descriptions of some solutions, such as 3D, 2D, and contour plots, give a clear view of their spatial-temporal development and physical inferences. These findings can improve the theoretical knowledge of dissipative solitons and give the basis of further evolution of acoustic metamaterials, nonlinear optics, and geophysical modeling. In this regard, our framework is an effective, versatile toolkit to solve high-dimensional nonlinear partial differential equations.