The highly nonlinear shallow water equation: local well-posedness, wave breaking data and non-existence of sech\(^2\) solutions

In the context of the initial data and an amplitude parameter \(\varepsilon \) $\epsilon$ , we establish a local existence result for a highly nonlinear shallow water equation on the real line. This result holds in the space \(H^k\) $H^k$ as long as \(k>5/2\) $k>5/2$ . Additionally, we illustrate that the threshold time for the occurrence of wave breaking in the surging type is on the order of \(\varepsilon ^{-1},\) ${\epsilon }^{-1},$ 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 \(sech^2\) $sec{h}^2$ .