Strong Confinement Limit for the Nonlinear Schrödinger Equation Constrained on a Curve

Abstract

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