×

A class of nonlinear Schrödinger equations with concentrated nonlinearity. (English) Zbl 0979.35130

The authors of this interesting paper consider the one-dimensional nonlinear Schrödinger equation with a nonlinearity concentrated in a finite number of points. The Schrödinger equation has the form: \[ i(\partial /\partial t)\psi (t)=-\triangle \psi (t)+\sum_{j=1}^n\alpha _j(t)\delta _{y_j}\psi (t), \quad \psi (0)=\psi _0, \] where \(\delta _{y_j}\) denotes the Dirac measure in \(y_j\) (\(j=1,2,\dots ,n\)) (\(n\) points on the real line), \(\alpha _j(t)\) (\(j=1,2,\dots ,n\)) are given functions. This problem describes an evolution with interactions concentrated in \(n\) points \(y_j\) of given time-dependent strengths \(\alpha _j(t)\) (\(j=1,2,\dots ,n\)). Detailed results on the local existence of the solutions in fractional Sobolev spaces \(H^p\) are shown. The conservation of the \(L^2-\)norm and the energy of solutions are proved. A global existence result for repulsive and weakly attractive interaction in the space \(H^1\) is shown as well. The blow-up phenomenon is investigated. This includes the existence of blow-up solutions for strongly attractive interaction.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B40 Asymptotic behavior of solutions to PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Adami, R.; Teta, A., A simple model of concentrated nonlinearity, Oper. Theory: Adv. Appl., 108, 183-189 (1999) · Zbl 0967.81010
[2] Adams, R., Sobolev Spaces (1975), Academic Press: Academic Press New York · Zbl 0314.46030
[3] Albeverio, S.; Gesztesy, F.; Högh-Krohn, R.; Holden, H., Solvable Models in Quantum Mechanics (1988), Springer-Verlag: Springer-Verlag New York
[4] Bulashenko, O. M.; Kochelap, V. A.; Bonilla, L. L., Coherent patterns and self-induced diffraction of electrons on a thin nonlinear layer, Phys. Rev. B, 54 (1996)
[5] Cazenave, T., An Introduction to Nonlinear Schrödinger Equations. An Introduction to Nonlinear Schrödinger Equations, Textos de Métodos Matematicos, 26 (1993), I.M.U.F.R.J: I.M.U.F.R.J Rio de Janeiro
[6] Cazenave, T., Blow up and Scattering in the Nonlinear Schrödinger Equation. Blow up and Scattering in the Nonlinear Schrödinger Equation, Textos de Métodos Matematicos, 30 (1996), I.M.U.F.R.J: I.M.U.F.R.J Rio de Janeiro
[7] Erdely, A., Tables of Integral Transform (1954), McGraw-Hill: McGraw-Hill New York
[8] Ginibre, J.; Velo, G., On a Class of Nonlinear Schrödinger Equations. I. The Cauchy Problem, General Case, J. Funct. Anal., 32, 1-32 (1979) · Zbl 0396.35028
[9] Gorenflo, R.; Vessella, S., Abel Integral Equations (1978), Springer-Verlag: Springer-Verlag Berlin/Heidelberg
[10] Jona-Lasinio, G.; Presilla, C.; Sjöstrand, J., On Schrödinger equations with concentrated nonlinearities, Ann. Phys., 240, 1-21 (1995) · Zbl 0820.34050
[11] Kato, T., On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré, Phys. Théorique, 46, 113-129 (1987) · Zbl 0632.35038
[12] Landman, M. J.; Papanicolaou, G. C.; Sulem, C.; Sulem, P. L., Rate of blowup for solutions of the nonlinear Schrödinger equation at critical dimension, Phys. Rev. A, 38, 3837-3843 (1988)
[13] Miller, R. K., Nonlinear Volterra Integral Equations (1971), Benjamin: Benjamin New York · Zbl 0209.14202
[14] Merle, F., Construction of solutions with exactly \(k\) blow-up points for the Schrödinger equation with critical nonlinearity, Comm. Math. Phys., 129, 223-240 (1990) · Zbl 0707.35021
[15] Malomed, B.; Azbel, M., Modulational instability of a wave scattered by a nonlinear center, Phys. Rev. B, 47, 16 (1993)
[16] Nier, F., The Dynamics of some Quantum Open System with Short-Range Nonlinearities (1998), Ecole Polytechnique · Zbl 0909.34052
[17] Rasmussen, J. J.; Rypdal, K., Blow-up in NLSE - A general review, Phys. Scripta, 33, 481-497 (1986) · Zbl 1063.35545
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.