S. Locke and D.E. Pelinovsky
On smooth and peaked traveling waves in a local model for shallow water waves
Abstract:
We introduce a new model equation for Stokes gravity waves based on
conformal transformations of Euler's equations. The local version of the model equation
is relevant for dynamics of shallow water waves. It allows us to characterize the
traveling periodic waves both in the case of smooth and peaked waves and to solve the
existence problem exactly, albeit not in elementary functions. Spectral stability of
smooth waves with respect to co-periodic perturbations is proven analytically based
on the exact count of eigenvalues in a constrained spectral problem.
Keywords:
Babenko equation; Stokes waves; existence and stability of nonlinear waves; period function;