Nonlinear effects in steady radiating waves: An exponential asymptotics approach

Abstract

An asymptotic study is made of nonlinear effects in steady radiating waves due to moving sources in dispersive media. The focus is on problems where the radiated waves have exponentially small amplitude with respect to a parameter μ << 1, as for instance free-surface waves due to a submerged body in the limit of low Froude number. In such settings, weakly nonlinear effects (controlled by the source strength ε) can be as important as linear propagation effects (controlled by μ), and computing the wave response for μ, ε << 1may require exponential (beyond-all-orders) asymptotics. This issue is discussed here using a simple model, namely, the forced Korteweg–de Vries (fKdV) equation where μ is the dispersion and ε is the nonlinearity parameter. The forcing term f(x) is assumed to be even and its Fourier transform ˆ f(k) to decay for k >> 1 like Akα exp(−βk), where A, α and β > 0 are free parameters. For this class of forcing profiles, the wave response hinges on beyond-all-orders asymptotics only if α > −1, and nonlinear effects differ fundamentally depending on whether α > 0, α = 0 or −1 < α < 0. Furthermore, the sign of the forcing amplitude parameter A is an important controlling factor of the nonlinear wave response. The asymptotic results compare favorably against direct numerical solutions of the fKdV equation for a wide range of μ and ε, in contrast to the linear wave response whose validity is rather limited.