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

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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 ψε in the limit ε→ 0, with an estimate of the approximation error, and derive a limiting nonlinear Schrödinger equation in dimension one. If the Cauchy data ψε₀ has a uniformly bounded energy, then it is a bounded sequence in H¹ and we show that the approximation is of order O(√ε). If we assume that ψε₀ is bounded in the graph norm of the Hamiltonian, then it is a bounded sequence in H² and we show that the approximation error is of order O(ε).