Very Singular Similarity Solutions and Hermitian Spectral Theory for Semilinear Odd-order PDEs

Abstract

Abstract. Asymptotic large-and short-time behaviour of solutions of the linear dispersion equationand its (2k + 1)th-order extensions are studied. Such a refined scattering is based on a "Hermitian" spectral theory for a pair {B, B * } of non self-adjoint rescaled operators and to its higher-order counterparts are presented. The goal is, by using various techniques, to show that there exists a countable sequence of critical exponents {p l = 1 + 3 l+1 , l = 0, 1, 2, ...} such that, at each p = p l , a p-branch of VSSs bifurcates from the corresponding eigenfunction ψ l of the linear operator B above.