Section: New Results

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

The preprint [58] 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 with an additional effective potential depending on the curvature.