13,702 research outputs found
Justification of the coupled-mode approximation for a nonlinear elliptic problem with a periodic potential
Coupled-mode systems are used in physical literature to simplify the
nonlinear Maxwell and Gross-Pitaevskii equations with a small periodic
potential and to approximate localized solutions called gap solitons by
analytical expressions involving hyperbolic functions. We justify the use of
the one-dimensional stationary coupled-mode system for a relevant elliptic
problem by employing the method of Lyapunov--Schmidt reductions in Fourier
space. In particular, existence of periodic/anti-periodic and decaying
solutions is proved and the error terms are controlled in suitable norms. The
use of multi-dimensional stationary coupled-mode systems is justified for
analysis of bifurcations of periodic/anti-periodic solutions in a small
multi-dimensional periodic potential.Comment: 18 pages, no figure
A non-conservative Harris ergodic theorem
We consider non-conservative positive semigroups and obtain necessary and
sufficient conditions for uniform exponential contraction in weighted total
variation norm. This ensures the existence of Perron eigenelements and provides
quantitative estimates of the spectral gap, complementing Krein-Rutman theorems
and generalizing probabilistic approaches. The proof is based on a
non-homogenous -transform of the semigroup and the construction of Lyapunov
functions for this latter. It exploits then the classical necessary and
sufficient conditions of Harris's theorem for conservative semigroups and
recent techniques developed for the study of absorbed Markov processes. We
apply these results to population dynamics. We obtain exponential convergence
of birth and death processes conditioned on survival to their quasi-stationary
distribution, as well as estimates on exponential relaxation to stationary
profiles in growth-fragmentation PDEs
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