Skip to main content
Log in

Mass-transfer instability of ground-states for Hamiltonian Schrödinger systems

  • Published:
Journal d'Analyse Mathématique Aims and scope

Abstract

We study generic semilinear Schrödinger systems which may be written in Hamiltonian form. In the presence of a single gauge invariance, the components of a solution may exchange mass between them while preserving the total mass. We exploit this feature to unravel new orbital instability results for ground-states. More precisely, we first derive a general instability criterion and then apply it to some well-known models arising in several physical contexts. In particular, this mass-transfer instability allows us to exhibit L2-subcritical unstable ground-states.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. P. Antonelli and R. M. Weishäupl, Asymptotic behavior of nonlinear Schrödinger systems with linear coupling, J. Hyperbolic Differ. Equ. 11 (2014), 159–183.

    Article  MathSciNet  MATH  Google Scholar 

  2. P. W. Bates and C. K. R. T. Jones, Invariant manifolds for semilinear partial differential equations, in Dynamics Reported, Vol. 2, Wiley, Chichester, 1989, pp. 1–38.

    MATH  Google Scholar 

  3. T. Cazenave, Semilinear Schrödinger Equations, American Mathematical Society, Providence, RI, 2003.

    Book  MATH  Google Scholar 

  4. M. Colin, L. Di Menza and J. C. Saut, Solitons in quadratic media, Nonlinearity 29 (2016), 1000–1035.

    Article  MathSciNet  MATH  Google Scholar 

  5. A. J. Corcho, S. Correia, F. Oliveira and J. D. Silva, On a nonlinear Schrödinger system arising in quadratic media, Commun. Math. Sci. 17 (2019), 969–987.

    Article  MathSciNet  MATH  Google Scholar 

  6. S. Correia, Ground-states for systems of M coupled semilinear Schrödinger equations with attraction-repulsion effects: characterization and perturbation results, Nonlinear Anal. 140 (2016), 112–129.

    Article  MathSciNet  MATH  Google Scholar 

  7. S. Correia, F. Oliveira and H. Tavares, Semitrivial vs. fully nontrivial ground states in cooperative cubic Schrödinger systems with d ≥ 3 equations, J. Funct. Anal. 271 (2016), 2247–2273.

    Article  MathSciNet  MATH  Google Scholar 

  8. J. M. Gonçalves Ribeiro, Instability of symmetric stationary states for some nonlinear Schrödinger equations with an external magnetic field, Ann. Inst. H. Poincaré Phys. Théor. 54 (1991), 403–433.

    MathSciNet  MATH  Google Scholar 

  9. M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal. 74 (1987), 160–197.

    Article  MathSciNet  MATH  Google Scholar 

  10. M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. II, J. Funct. Anal. 94 (1990), 308–348.

    Article  MathSciNet  MATH  Google Scholar 

  11. J. Jin, Z. Lin and C. Zeng, Dynamics near the solitary waves of the supercritical gKDV equations, J. Differential Equations 267 (2019), 7213–7262.

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Jüngel and R.-M. Weishäupl, Blow-up in two-component nonlinear Schrödinger systems with an external driven field, Math. Models Methods Appl. Sci. 23 (2013), 1699–1727.

    Article  MathSciNet  MATH  Google Scholar 

  13. T. Kapitula and K. Promislow, Spectral and Dynamical Stability of Nonlinear Waves, Springer, New York, 2013.

    Book  MATH  Google Scholar 

  14. T. Kapitula and B. Sandstede, Stability of bright solitary-wave solutions to perturbed nonlinear Schrödinger equations, Phys. D 124 (1998), 58–103.

    Article  MathSciNet  MATH  Google Scholar 

  15. E. Kirr, P. G. Kevrekidis and D. E. Pelinovsky, Symmetry-breaking bifurcation in the nonlinear Schrödinger equation with symmetric potentials, Comm. Math. Phys. 308 (2011), 795–844.

    Article  MathSciNet  MATH  Google Scholar 

  16. Y. S. Kivshar, A. A. Sukhorukov, E. A. Ostrovskaya, T. J. Alexander, O. Bang, S. M. Saltiel, C. B. Clausen and P. L. Christiansen, Multi-component optical solitary waves, Phys. A 288 (2000), 152–173.

    Article  Google Scholar 

  17. T.-C. Lin and J. Wei, Ground state of N coupled nonlinear Schrödinger equations in Rn, n ≤ 3, Comm. Math. Phys. 255 (2005), 629–653.

    Article  MathSciNet  MATH  Google Scholar 

  18. H. Liu, Z. Liu and J. Chang, Existence and uniqueness of positive solutions of nonlinear Schrödinger systems, Proc. Roy. Soc. Edinburgh Sect. A 145 (2015), 365–390.

    Article  MathSciNet  MATH  Google Scholar 

  19. B. A. Malomed, P. Drummond, H. He, A. Berntson, D. Anderson and M. Lisak, Spatiotemporal solitons in multidimensional optical media with a quadratic nonlinearity, Phys. Rev. E 56 (1997), 4725–4735.

    Article  Google Scholar 

  20. C. R. Menyuk, R. Schiek and L. Torner, Solitary waves due to χ(2):χ(2) cascading, J. Opt. B Quantum Semiclass. Opt. 11 (1994), 2434–2443.

    Google Scholar 

  21. D. Mihalache, D. Mazilu, L.-C. Crasovan and L. Torner, Stationary walking solitons in bulk quadratic nonlinear media, Optics Comm. 137 (1997), 113–117.

    Article  Google Scholar 

  22. K. Nakanishi and W. Schlag, Invariant Manifolds and Dispersive Hamiltonian Evolution Equations, European Mathematical Society (EMS), Zürich, 2011.

    Book  MATH  Google Scholar 

  23. F. Oliveira and A. Pastor, Onaschrödinger system arizing in nonlinear optics, Anal. Math. Phys. 11 (2021), Article no. 123.

  24. A. Pastor, On three-wave interaction Schrödinger systems with quadratic nonlinearities: global well-posedness and standing waves, Commun. Pure Appl. Anal. 18 (2019), 2217–2242.

    Article  MathSciNet  MATH  Google Scholar 

  25. R. A. Sammut, A. V. Buryak, and Y. S. Kivshar, Bright and dark solitary waves in the presence of third-harmonic generation, J. Opt. B Quantum Semiclass. Opt. 15 (1998), 1488–1496.

    MathSciNet  Google Scholar 

  26. J. Shatah and W. Strauss, Instability of nonlinear bound states, Comm. Math. Phys. 100 (1985), 173–190.

    Article  MathSciNet  MATH  Google Scholar 

  27. A. Stefanov, On the normalized ground states of second order PDE’s with mixed power non-linearities, Comm. Math. Phys. 369 (2019), 929–971.

    Article  MathSciNet  MATH  Google Scholar 

  28. E. Timmermans, Phase separation of Bose-Einstein condensates, Phys. Rev. Lett. 81 (1998), 5718–5721.

    Article  Google Scholar 

  29. S. Yin, Stability and instability of the standing waves for the Klein—Gordon—Zakharov system in one space dimension, Math. Methods Appl. Sci. 41 (2018), 4428–4447.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

S. Correia and J. D. Silva were partially supported by Fundação para a Ciência e Tecnologia, through CAMGSD, IST-ID (projects UIDB/04459/2020 and UIDP/04459/2020) and through the project NoDES (PTDC/MAT-PUR/1788/2020). F. Oliveira was partially supported by Fundação para a Ciência e Tecnologia, through the grant UID/MULTI/00491/2019 and through the project NoDES (PTDC/MAT-PUR/1788/2020).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Simão Correia.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Correia, S., Oliveira, F. & Silva, J.D. Mass-transfer instability of ground-states for Hamiltonian Schrödinger systems. JAMA 148, 681–710 (2022). https://doi.org/10.1007/s11854-022-0240-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11854-022-0240-5

Navigation