TY - JOUR

T1 - Eigenfunctions and very singular similarity solutions of odd-order nonlinear dispersion PDEs

T2 - Toward a "nonlinear Airy function" and others

AU - Fernandes, Ray S

AU - Galaktionov, Victor A

PY - 2012/8

Y1 - 2012/8

N2 - Asymptotic properties of nonlinear dispersion equations 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: 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) 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.

AB - Asymptotic properties of nonlinear dispersion equations 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: 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) 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.

UR - http://www.scopus.com/inward/record.url?scp=85027935649&partnerID=8YFLogxK

UR - http://dx.doi.org/10.1111/j.1467-9590.2012.00554.x

U2 - 10.1111/j.1467-9590.2012.00554.x

DO - 10.1111/j.1467-9590.2012.00554.x

M3 - Article

VL - 129

SP - 163

EP - 219

JO - Studies in Applied Mathematics

JF - Studies in Applied Mathematics

SN - 0022-2526

IS - 2

ER -