Analytical soliton solutions, bifurcation analysis, and physics-informed neural network validation of fluid wave dynamics in the (2+1) dimensional Boussinesq equation

This study develops a hybrid analytical and computational framework for the (2+1)-dimensional Boussinesq equation, which models weakly nonlinear, weakly dispersive shallow-water gravity waves relevant to wave resonance and oblique interactions in coastal and ocean engineering. Using the unified Riccati equation expansion (UREE) method, five exact traveling-wave solutions are constructed: a bright soliton, a singular soliton, two periodic waves, and a rational solution. Bifurcation analysis of the associated planar Hamiltonian system rigorously classifies these families and identifies a saddle–center bifurcation at the Riccati parameter μ = 0 , separating soliton, periodic, and rational regimes. Orbit topology is explicitly related to wave type through the Hamiltonian first integral. A physics-informed neural network (PINN), trained using the governing PDE residual with initial and boundary data, independently validates representative bright-soliton and periodic-wave solutions, achieving relative L 2 errors of approximately 16% and 14%, respectively, in reproducible training runs. The integration of UREE, dynamical-systems classification, and independent PINN validation has not previously been applied to this equation. This framework provides a physically interpretable approach for verifying exact solutions and analyzing nonlinear wave dynamics relevant to shallow-water resonance, undular bore formation, periodic wave trains, and related coastal engineering phenomena.