9 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
Coupled-mode equations and gap solitons in a two-dimensional nonlinear elliptic problem with a separable periodic potential
We address a two-dimensional nonlinear elliptic problem with a
finite-amplitude periodic potential. For a class of separable symmetric
potentials, we study the bifurcation of the first band gap in the spectrum of
the linear Schr\"{o}dinger operator and the relevant coupled-mode equations to
describe this bifurcation. The coupled-mode equations are derived by the
rigorous analysis based on the Fourier--Bloch decomposition and the Implicit
Function Theorem in the space of bounded continuous functions vanishing at
infinity. Persistence of reversible localized solutions, called gap solitons,
beyond the coupled-mode equations is proved under a non-degeneracy assumption
on the kernel of the linearization operator. Various branches of reversible
localized solutions are classified numerically in the framework of the
coupled-mode equations and convergence of the approximation error is verified.
Error estimates on the time-dependent solutions of the Gross--Pitaevskii
equation and the coupled-mode equations are obtained for a finite-time
interval.Comment: 32 pages, 16 figure
Coupled Mode Equations and Gap Solitons for the 2D Gross-Pitaevskii equation with a non-separable periodic potential
Gap solitons near a band edge of a spatially periodic nonlinear PDE can be
formally approximated by solutions of Coupled Mode Equations (CMEs). Here we
study this approximation for the case of the 2D Periodic Nonlinear
Schr\"{o}dinger / Gross-Pitaevskii Equation with a non-separable potential of
finite contrast. We show that unlike in the case of separable potentials [T.
Dohnal, D. Pelinovsky, and G. Schneider, J. Nonlin. Sci. {\bf 19}, 95--131
(2009)] the CME derivation has to be carried out in Bloch rather than physical
coordinates. Using the Lyapunov-Schmidt reduction we then give a rigorous
justification of the CMEs as an asymptotic model for reversible non-degenerate
gap solitons and even potentials and provide estimates for this
approximation. The results are confirmed by numerical examples including some
new families of CMEs and gap solitons absent for separable potentials.Comment: corrections of v.5: 1-assumption A.1 strengthened; 2-powers of
epsilon fixed in (4.25); 3-\hat{R}_j estimated in L_{s-2}^2 instead of L_s^2;
4-error estimate in Thm 4.9 fixed; 5- reversibility analysis in the
persistence step corrected, evenness of V added as an assumption for the
persistence ste
MODELING OF WAVE RESONANCES IN LOW-CONTRAST PHOTONIC CRYSTALS ∗
Abstract. Coupled-mode equations are derived from Maxwell equations for modeling of lowcontrast cubic-lattice photonic crystals in three spatial dimensions. Coupled-mode equations describe resonantly interacting Bloch waves in stop bands of the photonic crystal. We study the linear boundary-value problem for stationary transmission of four counter-propagating and two oblique waves on the plane. Well-posedness of the boundary-value problem is proved by using the method of separation of variables and generalized Fourier series. For applications in photonic optics, we compute integral invariants for transmission, reflection, and diffraction of resonant waves