PDE Seminar: Existence of line solitary waves and their instability in fifth-order Kadomtsev-Petviashvili equation by Giuseppe La Scala (Lund University/Scuola Superiore Meridionale)

Title: Existence of line solitary waves and their instability in fifth-order Kadomtsev-Petviashvili equation

Abstract: In Ming-Zhang-Zhang SIMA '12, it has been proved that in the critical surface tension case (namely, when the Bond number is close to 1/3), 3D capillary-gravity Water Wave equations with finite depth can be approximated with the fifth-order Kadomtsev-Petviashvili equation (KP) in the weak-transverse, long-wavelength regime. Such equation is locally well-posed, but global well-posedness is not clear. However, by symmetries we can provide some 1D global solutions called line solitary waves.

What about their stability? We show that for 1D longitudinal perturbations, their stability is decided by the (unknown) sign of a beyond-all-algebraic-order eigenvalue, arising from a symmetry-breaking phenomenon due to normal form approximations. In particular, instability is set if such eigenvalue is negative. And what if it is nonnegative? In such case, line solitary waves are transversally unstable with respect to both periodic and localized transverse perturbations.

Based on a joint work with Erik Wahlén (to appear).