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

Abstract

This paper is devoted to the cubic nonlinear Schrödinger equation in a two dimensional waveguide with shrinking cross section of order $\epsilon$. For a Cauchy data living essentially on the first mode of the transverse Laplacian, we provide a tensorial approximation of the solution ${\psi }^{\epsilon }$ in the limit $\epsilon \to 0$, with an estimate of the approximation error, and derive a limiting nonlinear Schrödinger equation in dimension one. If the Cauchy data ${\psi }_0^{\epsilon }$ has a uniformly bounded energy, then it is a bounded sequence in $H^1$ and we show that the approximation is of order $\mathcal{O}(\sqrt{\epsilon })$. If we assume that ${\psi }_0^{\epsilon }$ is bounded in the graph norm of the Hamiltonian, then it is a bounded sequence in $H^2$ and we show that the approximation error is of order $\mathcal{O}(\epsilon )$.

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