On water waves with a corner at the crest

A highly accurate technique for calculating the parameters of the Stokes 120° gravity wave on deep water is proposed. Tanaka's nonlinear transformation of the inverse plane is used to stretch the region near the wave crest for accelerating the convergence of the classical Michell series. Several families of new irregular self-intersecting profiles with a 120° angle at the crest are presented as well. They all degenerate into the Stokes 120° profile as numerical accuracy increases. These solutions seem to represent some kind of the so-called parasitic (or ghost) solutions, which emerge due to discretization of the original continuous problem. Probable physical relevance of such irregular solutions is discussed. The validity of the Stokes theorem about a 120° angle is tested numerically. No solutions with crest angles different from 120° were found. Hence, the Stokes 120° wave seems to be the unique gravity wave with a corner at the crest.

Wave packet pseudomodes of variable coefficient differential operators

The pseudospectra of non-selfadjoint linear ordinary differential operators with variable coefficients are considered. It is shown that when a certain winding number or twist condition is satisfied, closely related to Hörmander's commutator condition for partial differential equations, ϵ -pseudoeigenfunctions of such operators for exponentially small values of ϵ exist in the form of localized wave packets. In contrast to related results of Davies and of Dencker, Sjöstrand & Zworski, the symbol need not be smooth. Applications in fluid mechanics, non-hermitian quantum mechanics and other areas are presented with the aid of high-accuracy numerical computations.