Modeling nonlinear resonance: A modification to the stokes' perturbation expansion

Abstract

The Stokes' series is a small amplitude perturbation expansion for nonlinear, steadily translating waves of the form u(x - ct). We have developed a modification to the Stokes' perturbation expansion to cope with the type of resonance that occurs when two different wavenumbers have identical phase speeds. Although the nonlinear wave is smooth and bounded at the resonance, the traditional Stokes' expansion fails because of the often-encountered "small denominator" problem. The situation is rectified by adding the resonant harmonic into the expansion at lowest order. The coefficient of the resonant wave is determined at higher order. Near resonance is treated by expanding the dispersion parameter in terms of the amplitude. As an example, we have chosen the Korteweg de Vries equation with an additional fifth degree dispersion term. However, the method is applicable to the amplitude expansions of much more complicated problems, such as the double cnoidal waves of the Korteweg de Vries equation, the problem that motivated this study.