New Explicit Exact Solutions for the (1+1) Dimensional Generalized Shallow Water Wave Equation

The shallow water wave equation describes the propagation of waves in shallow water (where the water depth is small relative to the wavelength). It is widely used in various fields such as fluid dynamics, flood propagation, plasma physics, quantum field theory, and so on. This paper investigates the wave solution of the generalized shallow water wave equations using three effective methods: Kudryashov-expansion approach, the modified rational sine-cosine approach and the Hirota bilinear approach. We succeeded in obtaining kink solutions, singular kink solutions, periodic solutions, breathing solutions, bright solitons, and complex value solutions. Furthermore, the different wave solutions are depicted by constructing 2D and 3D diagrams to enhance the understanding and verification of our results. This paper is helpful to study the propagation law of fluctuations and the dynamic characteristics of wave changes, and has an important guiding role in port engineering, water resources dispatching, and the prevention of various natural disasters.