Strong confinement limit for the nonlinear Schr\"odinger equation constrained on a curve

TL;DR

This paper analyzes the behavior of the cubic nonlinear Schr"odinger equation in a thin waveguide as its cross section shrinks, deriving a one-dimensional limit equation and quantifying the approximation error.

Contribution

It provides a tensorial approximation of solutions and error estimates for the nonlinear Schr"odinger equation constrained on a shrinking waveguide, extending understanding of confinement limits.

Findings

Approximation error is O(√ε) for bounded energy initial data.

Approximation error is O(ε) when initial data is bounded in the graph norm.

Derived a one-dimensional limiting nonlinear Schr"odinger equation.

Abstract

This paper is devoted to the cubic nonlinear Schr\"odinger 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 Schr\"odinger equation in dimension one. If the Cauchy data $\psi^{\epsilon}_0$ has a uniformly bounded energy, then it is a bounded sequence in $H^1$ and we show that the approximation is of order $\mathcal O(\sqrt{\epsilon})$. If we assume that $\psi^{\epsilon}_0$ 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 $\mathcal O(\epsilon)$.