The emergence of rapid oscillations in a fifth-order KdV equation

Abstract

We consider the emergence of rapidly propagating, exponentially small-amplitude oscillations in the impulsively started fifth-order Korteweg--de Vries equation (KdV5), and demonstrate that these waves originate due to Stokes' phenomenon in a small-time boundary layer. We rescale the problem in time and construct an auxiliary linear PDE (lKdV5) that reproduces the relevant exponential asymptotic structure, and proceed by two main methods. Firstly, the Fourier transform solution of the auxiliary PDE is analysed via the method of steepest descents to extract the relevant asymptotic contributions and the locations of the Stokes curves. Secondly, we reproduce these results using a direct factorial-over-power analysis. In this analysis, we make use of the initial and far-field conditions to avoid performing the local asymptotic matching that is typically required by exponential asymptotic methods and is frequently impossible for nonlinear PDEs. A similar factorial-over-power analysis is then applied to the full nonlinear PDE, which cannot be studied using steepest descent methods. The results agree well with numerical simulations, demonstrating both the rapid propagation of non-decaying oscillations and transient ripples whose amplitudes decay in the far field, and revealing the complex Stokes structure (including both normal and higher-order Stokes curves) that governs their emergence. Such a complex Stokes structure in a temporal boundary layer is a novel phenomenon.