The NLS equation in dimension one with spatially concentrated nonlinearities: the pointlike limit

Abstract

In the present paper we study the following scaled nonlinear Schrödinger equation (NLS) in one space dimension: \[ i\frac{d}{dt} ψ^{\varepsilon}(t) =-Δψ^{\varepsilon}(t) + \frac{1}εV\left(\frac{x}ε\right)|ψ^{\varepsilon}(t)|^{2μ}ψ^{\varepsilon}(t) \quad \quad ε>0\ ,\quad V\in L^1(\mathbb{R},(1+|x|)dx) \cap L^\infty(\mathbb{R}) \ . \] This equation represents a nonlinear Schrödinger equation with a spatially concentrated nonlinearity. We show that in the limit $\epsilon \to 0$, the weak (integral) dynamics converges in $H^1(\mathbb{R})$ to the weak dynamics of the NLS with point-concentrated nonlinearity: \[ i\frac{d}{dt} ψ(t) =H_αψ(t) . \] where $H_{\alpha }$ is the laplacian with the nonlinear boundary condition at the origin ${\psi }^{\prime }(t,0+)-{\psi }^{\prime }(t,0-)=\alpha |\psi (t,0){|}^{2\mu }\psi (t,0)$ and $\alpha ={\int }_{\mathbb{R}}Vdx$. The convergence occurs for every $\mu \in {\mathbb{R}}^{+}$ if $V\ge 0$ and for every $\mu \in (0,1)$ otherwise. The same result holds true for a nonlinearity with an arbitrary number $N$ of concentration points

arXiv comment10 pages

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• A general review on the NLS equation with point-concentrated nonlinearity
• The Harmonic Oscillator Potential Perturbed by a Combination of Linear and Non-linear Dirac Delta Interactions with Application to Bose-Einstein Condensation
• Prescribed mass ground states for a doubly nonlinear Schrödinger equation in dimension one
• Well-posedness of the three-dimensional NLS equation with sphere-concentrated nonlinearity
• Microscopic derivation of a Schrödinger equation in dimension one with a nonlinear point interaction