Ostrovsky and Hunter's generic wave equation for weakly dispersive waves: matched asymptotic and pseudospectral study of the paraboloidal travelling waves (corner and near-corner waves)

Abstract

The wave equation $(u_{t} + u u_{x})_{x} - u = 0$ is a model for shallow water waves with Coriolis force, sound waves in a bubbly liquid and more generally “is the canonical asymptotic equation for genuinely nonlinear waves that are nondispersive as their wavelength tends to zero” in the words of Hunter [13]. This Ostrovsky–Hunter equation has steadily-translating, spatially periodic solutions which exist only when $c \leq c_{limit}$. The limiting wave (‘parabolic wave’) is exactly given by a piecewise quadratic polynomial in $x$ with a discontinuous slope at the crest. We show that near the limit, the travelling waves (‘paraboloidal waves’) can be approximated by matched asymptotic expansions: the inner solution rounds off the point while the outer solution, valid over most of the spatial domain, is to lowest order just the parabolic wave. In the opposite limit of small amplitude, we derive a Fourier-and-powers-of-amplitude expansion (‘Stokes' series’). We show that this is remarkably accurate even very close to the limiting wave and converges to the limiting wave for unit amplitude. We demonstrate also that the Fourier pseudospectral method gives first order convergence even for the slope-discontinuous parabolic wave.