Shallow Water Waves with Dispersion Triplet by the Complete Discriminant Approach

Abstract

This work investigates a class of nonlinear evolution equations that model shallow water wave dynamics and systematically recovers a rich variety of exact wave solutions, including solitary waves, shock waves, singular solitary waves, plane waves, and cnoidal waves. The methodology employed is the complete discriminant approach, a powerful analytical technique that leverages the structure of the polynomial nonlinearities within the governing equations. This approach enables the derivation of a complete spectrum of traveling wave solutions by classifying the roots of the associated algebraic equations based on the signs and multiplicities of the discriminant. By performing a rigorous caseby-case analysis, the study identifies the precise parametric conditions under which each wave type emerges, offering insight into the transition between different nonlinear wave phenomena. The analysis highlights how variations in physical parameters such as wave speed, dispersion coefficients, and nonlinearity strength govern the existence and shape of the obtained waveforms. The considered models and their solutions have broad relevance to coastal engineering, oceanography, and geophysical fluid dynamics, where understanding wave propagation, wave breaking, and pattern formation in shallow water environments is critical. The findings not only recover known wave structures in a unified framework but also reveal novel analytical forms under specific parametric regimes. This comprehensive treatment contributes to the theoretical understanding of shallow water wave dynamics and offers potential for further applications in numerical modeling and experimental validation in real-world shallow water systems.