Wilton ripples with high-order resonances in weakly nonlinear models

Speaker: Raymond Langer (UofT)

Abstract: Wilton ripples are a class of resonant periodic travelling waves that arise in nonlinear systems where different wavenumbers (k = 1 and k = N) may propagate at the same speed. We examine a family of weakly nonlinear partial differential equations which contains several important water wave equations. Previous work has focused on “triad resonances” where the resonant wavenumber is twice the base wavenumber, that is, N = 2. In this work, we focus on higher-order resonances, where the resonant wavenumber may be three or more times as large as the base wavenumber. In deriving explicit formulas for the coefficients in an asymptotic expansion, we discover several fascinating patterns, which are used to determine the lowest order at which the resonant wavenumber appears. We also highlight a recurring divisor, the size of which inversely correlates with the convergence of the asymptotics – when this divisor is zero, the asymptotics fail entirely. We expand on this behaviour using examples from the Kawahara and Benjamin equations.