Eigenfunctions and Very Singular Similarity Solutions of Odd‐Order Nonlinear Dispersion PDEs: Toward a “Nonlinear Airy Function” and Others

Abstract

Asymptotic properties of nonlinear dispersion equations with fixed exponents n > 0 and p > n+ 1, and their (2k+ 1)th‐order analogies are studied. The global in time similarity solutions, which lead to “nonlinear eigenfunctions” of the rescaled ordinary differential equations (ODEs), are constructed. The basic mathematical tools include a “homotopy‐deformation” approach, where the limit in the first equation in () turns out to be fruitful. At n= 0 the problem is reduced to the linear dispersion one: whose oscillatory fundamental solution via Airy’s classic function has been known since the nineteenth century. The corresponding Hermitian linear non‐self‐adjoint spectral theory giving a complete countable family of eigenfunctions was developed earlier in [1]. Various other nonlinear operator and numerical methods for () are also applied. As a key alternative, the “super‐nonlinear” limit , with the limit partial differential equation (PDE) admitting three almost “algebraically explicit” nonlinear eigenfunctions, is performed. For the second equation in (), very singular similarity solutions (VSSs) are constructed. In particular, a “nonlinear bifurcation” phenomenon at critical values {p=pl(n)}l≥0 of the absorption exponents is discussed.

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

“…In a forthcoming paper [15], these concepts and ideas will be applied and extended to PDEs with nonlinear dispersion (the NDEs) of the form…”