Stability of the standing waves of the concentrated NLSE in dimension two

Riccardo Adami; Raffaele Carlone; Michele Correggi; Lorenzo Tentarelli; Riccardo Adami; Raffaele Carlone; Michele Correggi; Lorenzo Tentarelli

doi:10.3934/mine.2021011

Mathematics in Engineering

2021, Volume 3, Issue 2: 1-15. doi: 10.3934/mine.2021011

Previous Article Next Article

Research article Special Issues

Stability of the standing waves of the concentrated NLSE in dimension two

1.
Politecnico di Torino, Dipartimento di Scienze Matematiche "G.L. Lagrange", Corso Duca degli Abruzzi, 24, 10129, Torino, Italy
2.
Università degli Studi di Napoli "Federico II", Dipartimento di Matematica e Applicazioni "R. Caccioppoli", MSA, via Cinthia, I-80126, Napoli, Italy
3.
Politecnico di Milano, Dipartimento di Matematica, Piazza Leonardo da Vinci, 32, 20133, Milano, Italy

Received: 10 January 2020 Accepted: 27 June 2020 Published: 13 July 2020
MSC : 35Q40, 35Q55

In this paper we will continue the analysis of two dimensional Schr?dinger equation with a fixed, pointwise, nonlinearity started in [2, 13]. In this model, the occurrence of a blow-up phenomenon has two peculiar features: the energy threshold under which all solutions blow up is strictly negative and coincides with the infimum of the energy of the standing waves; there is no critical power nonlinearity, i.e., for every power there exist blow-up solutions. Here we study the stability properties of stationary states to verify whether the anomalies mentioned before have any counterpart on the stability features.
- nonlinear Schr?dinger equation,
- point interactions,
- standing waves,
- orbital stability
Citation: Riccardo Adami, Raffaele Carlone, Michele Correggi, Lorenzo Tentarelli. Stability of the standing waves of the concentrated NLSE in dimension two[J]. Mathematics in Engineering, 2021, 3(2): 1-15. doi: 10.3934/mine.2021011

Related Papers:

Abstract

In this paper we will continue the analysis of two dimensional Schr?dinger equation with a fixed, pointwise, nonlinearity started in [2, 13]. In this model, the occurrence of a blow-up phenomenon has two peculiar features: the energy threshold under which all solutions blow up is strictly negative and coincides with the infimum of the energy of the standing waves; there is no critical power nonlinearity, i.e., for every power there exist blow-up solutions. Here we study the stability properties of stationary states to verify whether the anomalies mentioned before have any counterpart on the stability features.

References

[1]	Abramovitz M, Stegun IA (1965) Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables, New York: Dover.
[2]	Adami R, Carlone R, Correggi M, et al. (2020) Blow-up for the pointwise NLS in dimension two: Absence of critical power. J Differ Equations 269: 1-37.
[3]	Adami R, Dell'Antonio G, Figari R, et al. (2003) The Cauchy problem for the Schr?dinger equation in dimension three with concentrated nonlinearity. Ann I H Poincaré Anal Non Linéaire 20: 477-500.
[4]	Adami R, Dell'Antonio G, Figari R, et al. (2004) Blow-up solutions for the Schr?dinger equation in dimension three with a concentrated nonlinearity. Ann I H Poincaré Anal Non Linéaire 21: 121-137.
[5]	Adami R, Noja D (2013) Stability and symmetry-breaking bifurcation for the ground states of a NLS with a $\delta'$ interaction. Commun Math Phys 318: 247-289.
[6]	Adami R, Noja D, Ortoleva C (2013) Orbital and asymptotic stability for standing waves of a nonlinear Schr?dinger equation with concentrated nonlinearity in dimension three. J Math Phys 54: 013501.
[7]	Adami R, Noja D, Ortoleva C (2016) Asymptotic stability for standing waves of a NLS equation with subcritical concentrated nonlinearity in dimension three: neutral modes. Discrete Contin Dyn Syst 36: 5837-5879.
[8]	Adami R, Teta A (2001) A class of nonlinear Schr?dinger equations with concentrated nonlinearity. J Funct Anal 180: 148-175.
[9]	Azbel MY (1999) Quantum turbulence and resonant tunneling. Phys Rev B 59: 8049-8053.
[10]	Baro M, Neidhardt H, Rehberg J (2005) Current coupling of drift-diffusion models and Schr?dinger-Poisson systems: Dissipative hybrid models. SIAM J Math Anal 37: 941-981.
[11]	Bulashenko OM, Kochelap VA, Bonilla LL (1996) Coherent patterns and self-induced diffraction of electrons on a thin nonlinear layer. Phys Rev B 54: 1537-1540.
[12]	Cacciapuoti C, Carlone R, Noja D, et al. (2017) The 1-D Dirac equation with concentrated nonlinearity. SIAM J Math Anal 49: 2246-2268.
[13]	Carlone R, Correggi M, Tentarelli L (2019) Well-posedness of the two-dimensional nonlinear Schr?dinger equation with concentrated nonlinearity. Ann I H Poincaré Anal Non Linéaire 36: 257-294.
[14]	Carlone R, Figari R, Negulescu C (2017) The quantum beating and its numerical simulation. J Math Anal Appl 450: 1294-1316.
[15]	Carlone R, Finco D, Tentarelli L (2019) Nonlinear singular perturbations of the fractional Schr?dinger equation in dimension one. Nonlinearity 32: 3112-3143.
[16]	Carlone R, Fiorenza A, Tentarelli L (2017) The action of Volterra integral operators with highly singular kernels on H?lder continuous, Lebesgue and Sobolev functions. J Funct Anal 273: 1258-1294.
[17]	Cazenave T, Lions PL (1982) Orbital stability of standing waves for some nonlinear Schr?dinger equations. Commun Math Phys 85: 549-561.
[18]	Fukuizumi R, Ohta M, Ozawa T (2008) Nonlinear Schr?dinger equation with a point defect. Ann I H Poincaré Anal. Non Linéaire 25: 837-845.
[19]	Grillakis M, Shatah J, Strauss W (1987) Stability theory of solitary waves in the presence of symmetry. I. J Funct Anal 74: 160-197.
[20]	Jona-Lasinio G, Presilla C, Sjöstrand J (1995) On Schr?dinger equations with concentrated nonlinearities. Ann Phys 240: 1-21.
[21]	Merle F, Raphael P (2005) The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schr?dinger equation. Ann Math 161: 157-222.
[22]	Noris B, Tavares H, Verzini G (2019) Normalized solutions for nonlinear Schr?dinger systems on bounded domains. Nonlinearity 32: 1044.
[23]	Nier F (1998) The dynamics of some quantum open systems with short-range nonlinearities. Nonlinearity 11: 1127-1172.
[24]	Pierotti D, Verzini G (2017) Normalized bound states for the nonlinear Schr?dinger equation in bounded domains. Calc Var Partial Dif 56: 133.
[25]	Sánchez ó, Soler J (2004) Asymptotic decay estimates for the repulsive Schr?dinger-Poisson system. Math Method Appl Sci 27: 371-380.
[26]	Shatah J, Strauss WA (1985) Instability of nonlinear bound states. Commun Math Phys 100: 173-190.
[27]	Sivan Y, Fibich G, Efremidis NK, et al. (2008) Analytic theory of narrow lattice solitons. Nonlinearity 21: 509-536.
[28]	Stubbe J (1991) Global solutions and stable ground states of nonlinear Schr?dinger equations. Phys D 48: 259-272.
[29]	Sulem C, Sulem PL (1999) The Nonlinear Schr?dinger Equation, Self-Focusing and Wave Collapse, New York: Springer-Verlag.
[30]	Weinstein MI (1986) Lyapunov stability of ground states of nonlinear dispersive evolution equations. Commun Pure Appl Math 39: 51-68.
[31]	Woolard DL, Cui HL, Gelmont BL, et al. (2003) Advanced theory of instability in tunneling nanostructures. Int J High Speed Electron Syst 13: 1149-1253.

Reader Comments

Your name:*

Email:*
© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)