Eigenfunctions and Very Singular Similarity Solutions of Odd-Order Nonlinear Dispersion PDEs: Toward a 'Nonlinear Airy Function' and Others

摘要

Asymptotic properties of nonlinear dispersion equations u _t = (|u| ~nu)xxx and u _t = (|u| ~nu)xxx - |u|p-1u in R{double-struck} × R{double-struck} _+ (1) 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 (1) turns out to be fruitful. At n= 0 the problem is reduced to the linear dispersion one: ν _t = ν _(xxx) 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 (1) are also applied. As a key alternative, the "super-nonlinear" limit, with the limit partial differential equation (PDE) (sign _ν) t=ν _(xxx), in terms of the variable ν = |u| ~nu, admitting three almost "algebraically explicit" nonlinear eigenfunctions, is performed. For the second equation in (1), very singular similarity solutions (VSSs) are constructed. In particular, a "nonlinear bifurcation" phenomenon at critical values {p=p _l(n)} _l≥0 of the absorption exponents is discussed.