The highly nonlinear shallow water equation: local well-posedness, wave breaking data and non-existence of sech2 solutions

Abstract

In the context of the initial data and an amplitude parameter ε, we establish a local existence result for a highly nonlinear shallow water equation on the real line. This result holds in the space Hk as long as k > 5/2. Additionally, we illustrate that the threshold time for the occurrence of wave breaking in the surging type is on the order of ε−1, while plunging breakers do not manifest. Lastly, in accordance with ODE theory, it is demonstrated that there are no exact solitary wave solutions in the form of sech and sech2.