Tomáš Dohnal

Coupled Mode Equations and Gap Solitons for the 2D Gross-Pitaevskii equation with a non-separable periodic potential

Abstract 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 Schrodinger/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, G. Schneider, Coupled-mode equations and gap solitons in a two-dimensional nonlinear elliptic problem with a separable periodic potential, J. Nonlinear Sci. 19 (2009) 95–131] 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 provide H s estimates for this approximation. The results are confirmed by numerical examples, including some new families of CMEs and gap solitons absent for separable potentials.

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 Schrödinger 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 nondegeneracy 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 approximated by solutions of the coupled-mode equations are obtained for a finite-time interval.

Bloch-Wave Homogenization on Large Time Scales and Dispersive Effective Wave Equations

We investigate second order linear wave equations in periodic media, aiming at the derivation of effective equations in

Surface Gap Solitons at a Nonlinearity Interface

\mathbb{R}^n

Perfectly matched layers in photonics computations: 1D and 2D nonlinear coupled mode equations

,

Ground states of a nonlinear curl-curl problem in cylindrically symmetric media

n\in\{1,2,3\}

Perfectly matched layers for coupled nonlinear Schrödinger equations with mixed derivatives

. Standard homogenization theory provides, for the limit of a small periodicity length

Bifurcation of Nonlinear Bloch Waves from the Spectrum in the Gross–Pitaevskii Equation

\varepsilon>0

Dispersive homogenized models and coefficient formulas for waves in general periodic media

, an effective second order wave equation that describes solutions on time intervals

Dispersive effective equations for waves in heterogeneous media on large time scales

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