Skip to main content
SpringerLink
Log in

The linearly damped nonlinear Schrödinger equation with localized driving: spatiotemporal decay estimates and the emergence of extreme wave events

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

We prove spatiotemporal algebraically decaying estimates for the density of the solutions of the linearly damped nonlinear Schrödinger equation with localized driving, when supplemented with vanishing boundary conditions. Their derivation is made via a scheme, which incorporates suitable weighted Sobolev spaces and a time-weighted energy method. Numerical simulations examining the dynamics (in the presence of physically relevant examples of driver types and driving amplitude/linear loss regimes), showcase that the suggested decaying rates are proved relevant in describing the transient dynamics of the solutions, prior their decay: They support the emergence of waveforms possessing an algebraic space-time localization (reminiscent of the Peregrine soliton) as first events of the dynamics, but also effectively capture the space-time asymptotics of the numerical solutions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Fotopoulos, G., Frantzeskakis, D.J., Karachalios, N.I., Kevrekidis, P.G., Koukouloyannis, V., Vetas, K.: Communications in Nonlinear Science and Numerical Simulation. In: Extreme wave events for a nonlinear Schrödinger equation with linear damping and Gaussian driving, Vol. 82 (2020). https://doi.org/10.1016/j.cnsns.2019.105058

    Article  MathSciNet  Google Scholar 

  2. Karachalios, N.I., Kyriazopoulos, P., Vetas, K.: Excitation of Peregrine-type waveforms from vanishing initial conditions in the presence of periodic forcing. Z. Naturforsch. A. 74(5), 371–382 (2019)

    Article  Google Scholar 

  3. Peregrine, D.H.: Water waves, nonlinear Schrödinger equations and their solutions. J. Austral. Math. Soc. B 25, 16–43 (1983)

    Article  MathSciNet  Google Scholar 

  4. Cai, D., McLaughlin, D.W., McLaughlin, K.T.R.: The nonlinear Schrödinger equation as both a PDE and a dynamical system. In: Handbook of Dynamical Systems, vol. 2, pp. 599–675. North- Holland, Amsterdam (2002)

    MATH  Google Scholar 

  5. Bertola, M., Tovbis, A.: Universality for the focusing nonlinear Schrödinger equation at the gradient catastrophe point: rational breathers and poles of the Tritronquée solution to Painlevé. Commun. Pure Appl. Math. 66, 678–752 (2009)

    Article  Google Scholar 

  6. Grimshaw, R.H.J., Tovbis, A.: Rogue waves: analytical predictions. Proc. R. Soc. A 469, 20130094 (2013)

    Article  Google Scholar 

  7. Ghidaglia, J.M.: Finite dimensional behavior for the weakly damped driven Schrödinger equations. Ann. Inst. Henri Poincaré 5, 365–405 (1988)

    Article  MathSciNet  Google Scholar 

  8. Wang, X.: An energy equation for the weakly damped driven nonlinear Schrödinger equations and its application to their attractors. Phys. D 88, 167–175 (1995)

    Article  MathSciNet  Google Scholar 

  9. Goubet, O.: Regularity of the attractor for the weakly damped nonlinear Schrödinger equations. Appl. Anal. 60, 99–119 (1996)

    Article  MathSciNet  Google Scholar 

  10. Goubet, O.: Regularity of the attractor for Schrödinger equation. Appl. Math. Lett. 10, 57–59 (1997)

    Article  MathSciNet  Google Scholar 

  11. Goubet, O.: Regularity of the attractor for a weakly damped nonlinear Schrödinger equation in \({\mathbb{R}}^2\). Adv. Differ. Equ. 3, 337–360 (1998)

    MATH  Google Scholar 

  12. Goubet, O.: Global attractor for weakly damped nonlinear Schrödinger equations in \(L^2({\mathbb{R}})\). Nonlinear Anal. 71, 317–320 (2009)

    Article  MathSciNet  Google Scholar 

  13. Laurençot, P.: Long-time behaviour for weakly damped driven nonlinear Schrödinger equations in \({\mathbb{R}}^N\), \(N\le 3\). NoDEA Nonlinear Differ. Equ. Appl. 2, 357–369 (1995)

    Article  Google Scholar 

  14. Karachalios, N.I., Stavrakakis, N.M.: Global attractor for the weakly damped driven Schrödinger equation in \(H^2({\mathbb{R}})\). NoDEA Nonlinear Differ. Equ. Appl. 9, 347–360 (2002)

    Article  Google Scholar 

  15. Babin, A.V., Vishik, M.I.: Attractors for partial differential evolution equations in an unbounded domain. Proc. Roy. Soc. Edinb. Sect. A 116, 221–243 (1990)

    Article  MathSciNet  Google Scholar 

  16. Mielke, A., Schneider, G.: Attractors for modulation equations on unbounded domains-existence and comparison. Nonlinearity 8, 743–768 (1995)

    Article  MathSciNet  Google Scholar 

  17. Mielke, A.: The complex Ginzburg-Landau equation on large and unbounded domains: sharper bounds and attractors. Nonlinearity 10, 199–222 (1997)

    Article  MathSciNet  Google Scholar 

  18. Karachalios, N.I., Stavrakakis, N.M.: Existence of a global attractor for semilinear dissipative wave equations on \({\mathbb{R}}^N\). J. Differ. Equ. 157, 183–205 (1999)

    Article  Google Scholar 

  19. Iliyn, A.A.: Best constants in multiplicative inequalities for sup-norms. J. Lond. Math. Soc. 58, 84–96 (1998)

    Article  MathSciNet  Google Scholar 

  20. Bartuccelli, M., Deane, J., Zelik, S.: Asymptotic expansions and extremals for the critical Sobolev and Gagliardo-Nirenberg inequalities on a torus. Proc. Roy. Soc. Edinb. Sect. A 143, 445–482 (2013)

    Article  MathSciNet  Google Scholar 

  21. Dolbeault, J., Esteban, M.J., Laptev, A., Loss, M.: One-dimensional Gagliardo–Nirenberg–Sobolev inequalities: remarks on duality and flows. J. Lond. Math. Soc. 90, 525–550 (2014)

    Article  MathSciNet  Google Scholar 

  22. Sohr, H.: The Navier–Stokes Equations: An Elementary Functional Analytic Approach. Birkhäuser Advanced Texts, Birkhäuser (2000)

    MATH  Google Scholar 

  23. Kharif, C., Touboul, J.: Under which conditions the Benjamin-Feir instability may spawn an extreme wave event: a fully nonlinear approach. Eur. Phys. J. Spec. Top. 185, 159–168 (2010)

    Article  Google Scholar 

  24. Kharif, C., Kraenkel, R .A., Manna, M .A., Thomas, R.: The modulational instability in deep water under the action of wind and dissipation. J. Fluid Mech. 664, 138–149 (2010)

    Article  MathSciNet  Google Scholar 

  25. Slunyaev, A., Sergeeva, A., Pelinovsky, E.: Wave amplification in the framework of forced nonlinear Schrödinger equation: the rogue wave context. Phys. D 303, 18–27 (2015)

    Article  MathSciNet  Google Scholar 

  26. Onorato, M., Proment, D.: Approximate rogue wave solutions of the forced and damped nonlinear Schrödinger equation for water waves. Phys. Lett. A 376, 3057–3059 (2012)

    Article  Google Scholar 

  27. Brunetti, M., Marchiando, N., Berti, N., Kasparian, J.: Nonlinear fast growth of water waves under wind forcing. Phys. Lett. A 378, 1025–1030 (2014)

    Article  MathSciNet  Google Scholar 

  28. Chabchoub, A., Hoffmann, N., Branger, H., Kharif, C., Akhmediev, N.: Experiments on wind–perturbed rogue wave hydrodynamics using the Peregrine breather model. Phys. Fluids 25, 101704 (2013)

    Article  Google Scholar 

  29. Dostal, L., Hollm, M., Kreuzer, E.: Study on the behavior of weakly nonlinear water waves in the presence of random wind forcing. https://arxiv.org/abs/1909.11761

  30. Brezis, H.: Functional Analysis, Sobolev Spaces, and Partial Differential Equations. Springer, Berlin (2011)

    MATH  Google Scholar 

  31. Zeidler, E.: Nonlinear Functional Analysis and its Applications, Vol. II/A: Linear Monotone Operators. Springer, Berlin (1990)

    Book  Google Scholar 

  32. Simon, J.: Compact Sets in the Space \(L^p(0, T;B)\). Ann. Mat. Pura Appl. 146, 65–96 (1987)

    Article  MathSciNet  Google Scholar 

  33. Temam, R.: Infinite Dimensional Dynamical Systems in Mechanics and Physics. Springer, Berlin (1997)

    Book  Google Scholar 

  34. Kevrekidis, P.G.: The Discrete Nonlinear Schrödinger Equation: Mathematical Analysis, Numerical Computations and Physical Perspectives. Springer, Berlin (2009)

    Book  Google Scholar 

  35. Fotopoulos, G., Karachalios, N.I., Koukouloyannis, V., Vetas, K.: Collapse dynamics for the discrete nonlinear Schrödinger equation with gain and loss. Commun. Nonlinear Sci. Numer. Simul. 72, 213–231 (2019)

    Article  MathSciNet  Google Scholar 

  36. Trefethen, L.N.: Spectral Methods in MatLab. SIAM, Philadelphia (2000)

    Book  Google Scholar 

  37. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes 3rd Edition: The Art of Scientific Computing, 3rd Edition. Cambridge University Press, New York (2007)

    MATH  Google Scholar 

  38. Kraych, A., Suret, P., El, G., Randoux, S.: Nonlinear Evolution of the Locally Induced Modulational Instability in Fiber Optics. Phys. Rev. Lett. 122(5), 054101 (2019). https://doi.org/10.1103/PhysRevLett.122.054101

Download references

Acknowledgements

The authors acknowledge that this work was made possible by the NPRP Grant # [8-764-160] and NPRP Grant # [9-329-1-067] from the Qatar National Research Fund (a member of Qatar Foundation). The findings achieved herein are solely the responsibility of the authors.

Author information

Authors and Affiliations

  1. Department of Electrical Engineering and Computer Science, Khalifa University, P.O. Box 127788, Abu Dhabi, United Arab Emirates

    G. Fotopoulos

  2. Department of Mathematics, Statistics and Physics, College of Arts and Sciences, Qatar University, P.O. Box 2713, Doha, Qatar

    G. Fotopoulos, V. Koukouloyannis & K. Vetas

  3. Department of Mathematics, Laboratory of Applied Mathematics and Mathematical Modelling, University of the Aegean, 83200, Karlovassi, Samos, Greece

    N. I. Karachalios & V. Koukouloyannis

Authors
  1. G. Fotopoulos
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. I. Karachalios
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. V. Koukouloyannis
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. K. Vetas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to N. I. Karachalios.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fotopoulos, G., Karachalios, N.I., Koukouloyannis, V. et al. The linearly damped nonlinear Schrödinger equation with localized driving: spatiotemporal decay estimates and the emergence of extreme wave events. Z. Angew. Math. Phys. 71, 3 (2020). https://doi.org/10.1007/s00033-019-1223-y

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-019-1223-y

Mathematics Subject Classification

Keywords

Search

Navigation