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

This study is devoted to the cubic nonlinear Schrodinger 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 Schrodinger 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 \({\mathcal{O}(\varepsilon)}\).