ISSN:
1432-0916
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
,
Physics
Notes:
Abstract We establish existence of a dense set of non-linear eigenvalues,E, and exponentially localized eigenfunctions,u E , for some non-linear Schrödinger equations of the form $$Eu_E (x) = [( - \Delta + V(x))u_E ](x) + \lambda u_E (x)^3 ,$$ bifurcating off solutions of the linear equation with λ=0. The pointsx range over a lattice, ℤ d ,d=1,2,3,..., Δ is the finite difference Laplacian, andV(x) is a random potential. Such equations arise in localization theory and plasma physics. Our analysis is complicated by the circumstance that the linear operator −Δ+V(x) has dense point spectrum near the edges of its spectrum which leads to small divisor problems. We solve these problems by developing some novel bifurcation techniques. Our methods extend to non-linear wave equations with random coefficients.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01229204
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