Paper detail

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 $ψ^ε$ in the limit $ε\to 0$, with an estimate of the approximation error, and derive a limiting nonlinear Schrödinger equation in dimension one. If the Cauchy data $ψ^ε_0$ has a uniformly bounded energy, then it is a bounded sequence in $H^1$ and we show that the approximation is of order $\mathcal O(\sqrtε)$. If we assume that $ψ^ε_0$ 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 $\mathcal O(ε)$.