A general review on the NLS equation with point-concentrated nonlinearity

Lorenzo Tentarelli

doi:10.2478/caim-2023-0004

.blurhash-client-img { display: none !important; }

A general review on the NLS equation with point-concentrated nonlinearity

Communications in Applied and Industrial Mathematics

Volume 14 (2023): Issue 1 (January 2023)

By: Lorenzo Tentarelli

Open Access

|Oct 2023

Abstract

The paper presents a complete (to the best of the author’s knowledge) overview on the existing literature concerning the NLS equation with point-concentrated nonlinearity. Precisely, it mainly covers the following topics: definition of the model, weak and strong local well-posedness, global well-posedness, classification and stability (orbital and asymptotic) of the standing waves, blow-up analysis and derivation from the standard NLS equation with shrinking potentials. Also some related problem is mentioned.

References

F. Dalfovo, S. Giorgini, L. Pitaevskii, and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Reviews of Modern Physics, vol. 71, pp. 463–512, 1999.
Search in Google Scholar Back to article
B. Malomed, Nonlinear Schrödinger Equations, in Encyclopedia of Nonlinear Science (A. Scott, ed.), pp. 639–643, Routledge, 2005.
Search in Google Scholar Back to article
E. Fermi, Sul moto dei neutroni nelle sostanze idrogenate, Ricerca Scientifica, vol. 7, 1936.
Search in Google Scholar Back to article
O. Bulashenko, V. Kochelap, and L. Bonilla, Coherent patterns and self-induced diffraction of electrons on a thin nonlinear layer, Physical Review B, vol. 54, pp. 1537–1540, 1996.
Search in Google Scholar Back to article
G. J. Lasinio, C. Presilla, and J. Sjostrand, On Schrödinger equations with concentrated nonlinearities, Annals of Physics, vol. 240, pp. 1–21, 1995.
Search in Google Scholar Back to article
B. Malomed and M. Azbel, Modulational instability of a wave scattered by a nonlinear center, Physical Review B, vol. 47, pp. 10402–10406, 1993.
Search in Google Scholar Back to article
M. Molina and C. Bustamante, The attractive nonlinear delta-function potential, American Journal of Physics, vol. 70, pp. 67–70, 2002.
Search in Google Scholar Back to article
F.Nier, The dynamics of some quantum open systems with short-range nonlinearities, Nonlinearity, vol. 11, pp. 1127–1172, 1998.
Search in Google Scholar Back to article
C. Presilla, G. Jona-Lasinio, and F. Capasso, Nonlinear feedback oscillations in resonant tunneling through double barriers, Physical Review B, vol. 43, pp. 5200–5203, 1991.
Search in Google Scholar Back to article
A. Sukhorukov, Y. Kivshar, and O. Bang, Two-color nonlinear localized photonic modes, Physical Review E, vol. 60, pp. R41–R44, 1999.
Search in Google Scholar Back to article
A. Sukhorukov, Y. Kivshar, O. Bang, J. Rasmussen, and P. Christiansen, Nonlinearity and disorder: Classification and stability of nonlinear impurity modes, Physical Review E, vol. 63, pp. 366011–3660118, 2001.
Search in Google Scholar Back to article
P. Yeh, Optical Waves in Layered Media. Wiley, 2005.
Search in Google Scholar Back to article
N. Dror and B. Malomed, Solitons supported by localized nonlinearities in periodic media, Physical Review A, vol. 83, Paper No. 033828, 2011.
Search in Google Scholar Back to article
K. Li, P. Kevrekidis, B. Malomed, and D. Frantzeskakis, Transfer and scattering of wave packets by a nonlinear trap, Physical Review E, vol. 84, Paper No. 056609, pp. 1103–1128, 2011.
Search in Google Scholar Back to article
H. Sakaguchi and B. Malomed, Singular solitons, Physical Review E, vol. 101, Paper No. 012211, 2020.
Search in Google Scholar Back to article
E. Shamriz, Z. Chen, B. Malomed, and H. Sakaguchi, Singular mean-field states: A brief review of recent results, Condensed Matter, vol. 5, Paper No. 20, 2020.
Search in Google Scholar Back to article
R. Adami and D. Noja, Existence of dynamics for a 1D NLS equation perturbed with a generalized point defect, Journal of Physics. A. Mathematical and Theoretical, vol. 42, Paper No. 495302, 2009.
Search in Google Scholar Back to article
R. Adami and D. Noja, Stability and symmetry-breaking bifurcation for the ground states of a NLS with a δ¹ interaction, Communications in Mathematical Physics, vol. 318, pp. 247–289, 2013.
Search in Google Scholar Back to article
R. Adami, D. Noja, and N. Visciglia, Constrained energy minimization and ground states for NLS with point defects, Discrete and Continuous Dynamical Systems - Series B, vol. 18, pp. 1155–1188, 2013.
Search in Google Scholar Back to article
K. Datchev and H. J, Fast soliton scattering by attractive delta impurities, Communications in Partial Differential Equations, vol. 34, pp. 1074–1113, 2009.
Search in Google Scholar Back to article
R. Fukuizumi and L. Jeanjean, Stability of standing waves for a nonlinear Schrödinger equation with a repulsive Dirac delta potential, Discrete and Continuous Dynamical Systems, vol. 21, pp. 121–136, 2008.
Search in Google Scholar Back to article
R. Fukuizumi, M. Otha, and T. Ozawa, Nonlinear Schrödinger equation with a point defect, Annales de l’Institut Henri Poincaré C, Analyse non linéaire, vol. 25, pp. 837–845, 2008.
Search in Google Scholar Back to article
R. Goodman, P. Holmes, and M. Weinstein, Strong NLS soliton-defect interactions, Physica D: Nonlinear Phenomena, vol. 192, pp. 215–248, 2004.
Search in Google Scholar Back to article
J.Holmer, J. Marzuola, and M. Zworski, Fast soliton scattering by delta impurities, Communications in Mathematical Physics, vol. 274, pp. 187–216, 2007.
Search in Google Scholar Back to article
S. L. Coz, R. Fukuizumi, G. Fibich, B. Ksherim, and Y. Sivan, Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential, Physica D: Nonlinear Phenomena, vol. 237, pp. 1103–1128, 2008.
Search in Google Scholar Back to article
J. Murphy and K. Nakanishi, Failure of scattering to solitary waves for long-range nonlinear Schrödinger equations, Discrete and Continuous Dynamical Systems, vol. 41, pp. 1507–1517, 2021.
Search in Google Scholar Back to article
R. Adami, F. Boni, R. Carlone, and L. Tentarelli, Ground states for the planar NLSE with a point defect as minimizers of the constrained energy, Calculus of Variations and Partial Differential Equations, vol. 61, Paper No. 195, 2022.
Search in Google Scholar Back to article
R. Adami, F. Boni, R. Carlone, and L. Tentarelli, Existence, structure, and robustness of ground states of a NLSE in 3D with a point defect, Journal of Mathematical Physics, vol. 63, Paper No. 071501, 2022.
Search in Google Scholar Back to article
C. Cacciapuoti, D. Finco, and D. Noja, Well posedness of the nonlinear Schrödinger equation with isolated singularities, Journal of Differential Equations, vol. 305, pp. 288–318, 2021.
Search in Google Scholar Back to article
C. Cacciapuoti, D. Finco, and D. Noja, Failure of scattering for the NLSE with a point interaction in dimension two and three, arXiv:2212.14216 [math-ph], 2023.
Search in Google Scholar Back to article
N. Fukaya, V. Georgiev, and M. Ikeda, On stability and instability of standing waves for 2d-nonlinear Schrödinger equations with point interaction, Journal of Differential Equations, vol. 321, pp. 258–295, 2022.
Search in Google Scholar Back to article
D. Finco and D. Noja, Blow-up and instability of standing waves for the NLS with a point interaction in dimension two, Zeitschrift für angewandte Mathematik und Physik, vol. 74, Paper No. 162, 2023.
Search in Google Scholar Back to article
T. Fülöp and I. Tsutsui, A free particle on a circle with point interaction, Physics Letters A, vol. 264, pp. 366–374, 2000.
Search in Google Scholar Back to article
F. Boni and R. Carlone, NLS ground states on the half-line with point interactions, Nonlinear Differential Equations and Applications NoDEA, vol. 30, Paper No. 51, 2023.
Search in Google Scholar Back to article
R. Adami, F. Boni, and S. Dovetta, Competing nonlinearities in NLS equations as source of threshold phenomena on star graphs, Journal of Functional Analysis, vol. 283, Paper No. 109483, 2022.
Search in Google Scholar Back to article
F. Boni and S. Dovetta, Prescribed mass ground states for a doubly nonlinear Schrödinger equation in dimension one, Journal of Mathematical Analysis and Applications, vol. 496, Paper No. 124797, 2021.
Search in Google Scholar Back to article
F. Boni and S. Dovetta, Doubly nonlinear Schrödinger ground states on metric graphs, Nonlinearity, vol. 35, pp. 3283–3323, 2022.
Search in Google Scholar Back to article
S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden, Solvable Models in Quantum Mechanics. Springer-Verlag, 1988.
Search in Google Scholar Back to article
R. Adami, A Class of Schrödinger Equations with Concentrated Nonlinearity. PhD thesis, Department of Mathematics “Guido Castelnuovo”, 2000.
Search in Google Scholar Back to article
M. Abramovitz and I. Stegun, Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables. Dover, 1965.
Search in Google Scholar Back to article
M. Reed and B. Simon, Methods of modern mathematical physics. II. Fourier analysis, self-adjointness. Academic Press, 1975.
Search in Google Scholar Back to article
S. Albeverio, Z. Brzežniak, and L. Dabrowski, Time-dependent propagator with point interaction, Journal of Physics. A. Mathematical and General, vol. 27, pp. 4933–4943, 1994.
Search in Google Scholar Back to article
R. Adami and A. Teta, A class of nonlinear Schrödinger equations with concentrated nonlinearity, Journal of Functional Analysis, vol. 180, pp. 148–175, 2001.
Search in Google Scholar Back to article
C. Carlone, M. Correggi, and L. Tentarelli, Well-posedness of the two-dimensional nonlinear Schrödinger equation with concentrated nonlinearity, Annales de l’Institut Henri Poincaré C, Analyse non linéaire, vol. 36, pp. 257–294, 2019.
Search in Google Scholar Back to article
R. Adami, G. Dell’Antonio, R. Figari, and A. Teta, The Cauchy problem for the Schrödinger equation in dimension three with concentrated nonlinearity, Annales de l’Institut Henri Poincaré C, Analyse non linéaire, vol. 20, pp. 477–500, 2003.
Search in Google Scholar Back to article
C. Carlone, A. Fiorenza, and L. Tentarelli, The action of Volterra integral operators with highly singular kernels on Hölder continuous, Lebesgue and Sobolev functions, Journal of Functional Analysis, vol. 273, pp. 1258–1294, 2017.
Search in Google Scholar Back to article
C. Carlone, D. Finco, and L. Tentarelli, Nonlinear singular perturbations of the fractional Schrödinger equation in dimension one, Nonlinearity, vol. 32, pp. 3112–3143, 2019.
Search in Google Scholar Back to article
C. Cacciapuoti, D. Finco, D. Noja, and A. Teta, The point-like limit for a NLS equation with concentrated nonlinearity in dimension three, Journal of Functional Analysis, vol. 273, pp. 1762–1809, 2017.
Search in Google Scholar Back to article
R. Adami, R. Carlone, M. Correggi, and L. Tentarelli, Stability of the standing waves of the concentrated NLSE in dimension two, Mathematics in Engineering, vol. 3, pp. 1–15, 2021.
Search in Google Scholar Back to article
R. Adami, D. Noja, and C. Ortoleva, Orbital and asymptotic stability for standing waves of a nonlinear Schrödinger equation with concentrated nonlinearity in dimension three, Journal of Mathematical Physics, vol. 54, Paper No. 013501, 2013.
Search in Google Scholar Back to article
M. Grillakis, J. Shatah, and W. Strauss, Stability theory of solitary waves in the presence of symmetry. I, Journal of Functional Analysis, vol. 74, pp. 160–197, 1987.
Search in Google Scholar Back to article
V. Buslaev, A. Komech, E. Kopylova, and D. Stuart, On asymptotic stability of solitary waves in Schrödinger equation coupled to nonlinear oscillator, Communications in Partial Differential Equations, vol. 33, pp. 669–705, 2008.
Search in Google Scholar Back to article
A. Komech, E. Kopylova, and D. Stuart, On asymptotic stability of solitons in a nonlinear Schrödinger equation, Communications on Pure and Applied Analysis, vol. 11, pp. 1063–1079, 2012.
Search in Google Scholar Back to article
R. Adami, D. Noja, and C. Ortoleva, Asymptotic stability for standing waves of a NLS equation with subcritical concentrated nonlinearity in dimension three: neutral modes, Discrete and Continuous Dynamical Systems - Series B, vol. 36, pp. 5837–5879, 2016.
Search in Google Scholar Back to article
R. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, Journal of Mathematical Physics, vol. 18, pp. 1794–179, 1977.
Search in Google Scholar Back to article
R. Adami, R. Carlone, M. Correggi, and L. Tentarelli, Blow-up for the pointwise NLS in dimension two: absence of critical power, Journal of Differential Equations, vol. 269, pp. 1–37, 2020.
Search in Google Scholar Back to article
R. Adami, G. Dell’Antonio, R. Figari, and A. Teta, Blow-up solutions for the Schrödinger equation in dimension three with a concentrated nonlinearity, Annales de l’Institut Henri Poincaré C, Analyse non linéaire, vol. 21, pp. 121–137, 2004.
Search in Google Scholar Back to article
R. Adami and A. Teta, A simple model of concentrated nonlinearity, Operator Theory: Advances and Applications, vol. 108, pp. 183–189, 1999.
Search in Google Scholar Back to article
J.Holmer and C.Liu, Blow-up for the 1d nonlinear Schrödinger equation with point nonlinearity I: Basic theory, Journal of Mathematical Analysis and Applications, vol. 483, Paper No. 123522, 2020.
Search in Google Scholar Back to article
J.Holmer and C.Liu, Blow-up for the 1d nonlinear Schrödinger equation with point nonlinearity II: supercritical blow-up profiles, Communications on Pure and Applied Analysis, vol. 20, pp. 215–242, 2021.
Search in Google Scholar Back to article
R. Adami, R. Fukuizumi, and J. Holmer, Scattering for the L²-supercritical point NLS, Transactions of the American Mathematical Society, vol. 374, pp. 35–60, 2021.
Search in Google Scholar Back to article
T. Cazenave, Semilinear Schrödinger equations. American Mathematical Society, 2003.
Search in Google Scholar Back to article
T. Tao, Nonlinear dispersive equations. Local and global analysis. American Mathematical Society, 2006.
Search in Google Scholar Back to article
C. Cacciapuoti, D. Finco, D. Noja, and A. Teta, The NLS equation in dimension one with spatially concentrated nonlinearities: the pointlike limit, Letters in Mathematical Physics, vol. 104, pp. 1557–1570, 2014.
Search in Google Scholar Back to article
A. Michelangeli, A. Ottolini, and R. Scandone, Fractional powers and singular perturbations of quantum differential Hamiltonians, Journal of Mathematical Physics, vol. 59, Paper No. 072106, 2018.
Search in Google Scholar Back to article
J. Antoine, F. Gesztesy, and J. Shabani, Exactly solvable models of sphere interactions in quantum mechanics, Journal of Physics. A. Mathematical and General, vol. 20, pp. 3687–3712, 1987.
Search in Google Scholar Back to article
J. Behrndt, P. Exner, and V. Lotoreichik, Schrödinger operators with δ-interactions supported on conical surfaces, Journal of Physics. A. Mathematical and Theoretical, vol. 47, Paper No. 355202, 2014.
Search in Google Scholar Back to article
J. Behrndt, P. Exner, M. Holzmann, and V. Lotoreichik, Approximation of Schrödinger operators with δ-interactions supported on hypersurfaces, Mathematische Nachrichten, vol. 290, pp. 1215–1248, 2017.
Search in Google Scholar Back to article
J. Behrndt, M. Langer, and V. Lotoreichik, Schrödinger operators with δ and δ¹-potentials supported on hypersurfaces, Annales Henri Poincaré, vol. 14, pp. 385–423, 2013.
Search in Google Scholar Back to article
D. Finco, L. Tentarelli, and A. Teta, Well-posedness of the three-dimensional NLS equation with sphere-concentrated nonlinearity, arXiv:2209.13504 [math.AP], 2022.
Search in Google Scholar Back to article
C. Carlone, M. Correggi, M. Falconi, and M. Olivieri, Emergence of time-dependent point interactions in polaron models, SIAM Journal on Mathematical Analysis, vol. 53, pp. 4657–4691, 2021.
Search in Google Scholar Back to article
M. Correggi, M. Falconi, and M. Olivieri, Quasi-classical dynamics, Journal fo the European Mathematical Society, vol. 25, pp. 731–783, 2023.
Search in Google Scholar Back to article
C. Kittel, Introduction to Solid State Physics - 8th edn. Wiley, 2004.
Search in Google Scholar Back to article
T. Hmidi, A. Mantine, and F. Nier, Time-dependent delta-interactions for 1d Schrödinger Hamiltonians, Mathematical Physics, Analysis and Geometry, vol. 13, pp. 83–103, 2010.
Search in Google Scholar Back to article
W. Borrelli, R. Carlone, and L. Tentarelli, Complete ionization for a non-autonomous point interaction model in d “ 2, Communications in Mathematical Physics, vol. 395, pp. 963–1005, 2022.
Search in Google Scholar Back to article
M. Sayapova and D. Yafaev, The evolution operator for time-dependent potentials of zero radius, Akademiya Nauk SSSR. Trudy Matematicheskogo Instituta imeni V. A. Steklova, vol. 159, pp. 167–174, 1983.
Search in Google Scholar Back to article
O. Costin, R. Costin, J. Lebowitz, and A. Rokhlenko, Evolution of a model quantum system under time periodic forcing: conditions for complete ionization, Communications in Mathematical Physics, vol. 221, pp. 1–26, 2001.
Search in Google Scholar Back to article
O. Costin, R. Costin, and J. Lebowitz, Nonperturbative time dependent solution of a simple ionization model, Communications in Mathematical Physics, vol. 361, pp. 217–238, 2018.
Search in Google Scholar Back to article
M. Correggi, G. Dell’Antonio, R. Figari, and A. Mantile, Ionization for three dimensional time-dependent point interactions, Communications in Mathematical Physics, vol. 257, pp. 169–192, 2005.
Search in Google Scholar Back to article
G. Dell’Antonio, R. Figati, and A. Teta, The Schrödinger equation with moving point interactions in three dimensions, in Stochastic processes, physics and geometry: new interplays, I (F. Gesztesy, H. Holden, J. Jost, S. Paycha, M. Röckner, and S. Scarlatti, eds.), pp. 99–113, American Mathematical Society, 2000.
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/caim-2023-0004 | Journal eISSN: 2038-0909

Journal RSS Feed

Language: English

Page range: 62 - 84

Submitted on: Apr 18, 2023

Accepted on: Jul 28, 2023

Published on: Oct 13, 2023

Published by: Italian Society for Applied and Industrial Mathemathics

In partnership with: Paradigm Publishing Services

Publication frequency: 1 issue per year

Keywords:

NLS equation,

concentrated nonlinearity,

blow-up,

Related subjects:

Numerical and computational mathematics,

Applied mathematics

© 2023 Lorenzo Tentarelli, published by Italian Society for Applied and Industrial Mathemathics
This work is licensed under the Creative Commons Attribution 4.0 License.

Volume 14 (2023): Issue 1 (January 2023)