Wilton Ripples with High-Order Resonances in Weakly Nonlinear Models

Resonant periodic traveling waves (Wilton ripples) are examined asymptotically for a family of weakly nonlinear partial differential equations. Wilton ripple resonances can occur between pairs of wavenumbers, here labeled k = 1 and N . Typical studies consider N = 2 , the triad resonance, but this work examines N > 2 , denoted “high-order resonance.” We present explicit formulas for the coefficients in the asymptotic series expansions and answer previously unaddressed questions including what modes are present at each order in the asymptotic expansion and at what order we can expect a non-zero Wilton ripple. The character of the solutions is presented using the example of the Kawahara equation. Finally, we comment on the factors which are indicative of convergence for the asymptotic expansions, and present an example where the series degenerates in the Benjamin equation.