Strong confinement limit for the nonlinear Schrödinger equation constrained on a curve

This paper is devoted to the cubic nonlinear Schrödinger equation in a two dimensional waveguide with shrinking cross section of order ε. For a Cauchy data living essentially on the first mode of the transverse Laplacian, we provide a tensorial approximation of the solution ψε in the limit ε → 0, with an estimate of the approximation error, and derive a limiting nonlinear Schrödinger equation in dimension one. If the Cauchy data ψε 0 has a uniformly bounded energy, then it is a bounded sequence in H 1 and we show that the approximation is of order O(√ε). If we assume that ψε 0 is bounded in the graph norm of the Hamiltonian, then it is a bounded sequence in H and we show that the approximation error is of order O(ε). 1 Motivation and results 1.1 Motivation The Dirichlet realization of the Laplacian on tubes of the Euclidean space plays an important role in the physical description of nanostructures. In the last twenty years, many papers were concerned by the influence of the geometry of the tube (curvature, torsion) on the spectrum. For instance, in [12], Duclos and Exner proved that bending a waveguide in dimension two and three always induces the existence of discrete spectrum below the essential spectrum (see also [10]). Another question of interest in their paper is the limit when the cross section shrinks to a point. In particular they prove that, in some sense, the Dirichlet Laplacian on a bidimensional tube, with cross section (−ε, ε) is well approximated by Schrödinger operator −∂ x1 − κ(x1) 4 − 1 ε2 ∂ x2 , acting on L2(R × (−1, 1), dx1 dx2) and where κ denotes the curvature of the center line of the tube. Such approximations have been recently considered in [15] or in presence of magnetic fields [14] through a convergence of resolvent method. Concerning this kind of results, one may refer to the memoir by Wachsmuth and Teufel [18] where dynamical problems are analyzed in the spirit of adiabatic reductions. In the present paper, we will consider the time dependent Schrödinger equation with a cubic non linearity in a waveguide and we would especially like to determine if the adiabatic reduction available in the linear framework can be used to reduce the dimension of the non linear equation and provide an effective dynamics in dimension one. The derivation of nonlinear quantum models in reduced dimensions has been the object of several works in the past years. For the modeling of the dynamics of electrons in nanostructures, the