Symmetric waves are traveling waves of some shallow water scalar equations

Abstract

Following a straightforward proof for symmetric solutions to be traveling waves by Pei (Exponential decay and symmetry of solitary waves to Degasperis-Procesi equation. Journal of Differential Equations. 2020;269(10):7730-7749), we prove that classical symmetric solutions of the highly nonlinear shallow water equation recently derived by Quirchmayr (A new highly nonlinear shallow water wave equation. Journal of Evolution Equations. 2016;16(3):539-556) are indeed traveling waves, with further information on their steady structures. We also provide a simple proof that symmetric waves are traveling waves to the free surface evolution equation of moderate amplitude waves in shallow water.