Title
Existence of line solitary waves and their instability in
fifth-order Kadomtsev-Petviashvili equation
Abstract
In Ming-Zhang-Zhang SIAM '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 on arXiv.
