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

An upper bound for validity limits of asymptotic analytical approaches based on normal form theory

Perturbation methods are routinely used in all fields of applied mathematics where analytical solutions for nonlinear dynamical systems are searched. Among them, normal form theory provides a reliable method for systematically simplifying dynamical systems via nonlinear change of coordinates, and is also used in a mechanical context to define Nonlinear Normal Modes (NNMs). The main recognized drawback of perturbation methods is the absence of a criterion establishing their range of validity in terms of amplitude. In this paper, we propose a method to obtain upper bounds for amplitudes of changes of variables in normal form transformations. The criterion is tested on simple mechanical systems with one and two degrees-of-freedom, and for complex as well as real normal form. Its behavior with increasing order in the normal transform is established, and comparisons are drawn between exact solutions and normal form computations for increasing levels of amplitudes. The results clearly establish that the criterion gives an upper bound for validity limit of normal transforms

Reflection ideals and mappings between generic submanifolds in complex space

International audienceResults on finite determination and convergence of formal mappings between smooth generic submanifolds in \C^N are established in this paper. The finite determination result gives sufficient conditions to guarantee that a formal map is uniquely determined by its jet, of a preassigned order, at a point. Convergence of formal mappings for real-analytic generic submanifolds under appropriate assumptions is proved, and natural geometric conditions are given to assure that if two germs of such submanifolds are formally equivalent, then they are necessarily biholomorphically equivalent. It is also shown that if two real-algebraic hypersurfaces in \C^N are biholomorphically equivalent, then they are algebraically equivalent. All the results are first proved in the more general context of ``reflection ideals" associated to formal mappings between formal as well as real-analytic and real-algebraic manifolds

Extended Crystal PDE's

In this paper we show that between PDE's and crystallographic groups there is an unforeseen relation. In fact we prove that integral bordism groups of PDE's can be considered extensions of crystallographic subgroups. In this respect we can consider PDE's as {\em extended crystals}. Then an algebraic-topological obstruction ({\em crystal obstruction}), characterizing existence of global smooth solutions for smooth boundary value problems, is obtained. Applications of this new theory to the Ricci-flow equation and Navier-Stokes equation are given that solve some well-known fundamental problems. These results, are also extended to singular PDE's, introducing ({\em extended crystal singular PDE's}). An application to singular MHD-PDE's, is given following some our previous results on such equations, and showing existence of (finite times stable smooth) global solutions crossing critical nuclear energy production zone