AIMS Mathematics, 2020, 5(5): 4596-4612. doi: 10.3934/math.2020295.

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Instability of standing waves for the inhomogeneous Gross-Pitaevskii equation

1 Department of Basic Teaching and Research, Qinghai University, Xining, 810016, China
2 Department of Mathematics, Northwest Normal University, Lanzhou, 730070, China

In this paper, we consider the instability of standing waves for an inhomogeneous Gross-Pitaevskii equation \[ i\psi_t +\Delta \psi -a^2|x|^2\psi +|x|^{-b}|\psi|^{p}\psi=0. \] This equation arises in the description of nonlinear waves such as propagation of a laser beam in the optical fiber. We firstly proved that there exists $\omega_*>0$ such that for all $\omega>\omega_*$, the standing wave $\psi(t,x)=e^{i\omega t}u_\omega(x)$ is unstable. Then, we deduce that if $\partial_\lambda^2S_\omega(u_\omega^\lambda)|_{\lambda=1}\leq 0$, the ground state standing wave $e^{i\omega t}u_\omega(x)$ is strongly unstable by blow-up, where $u_\omega^\lambda(x)=\lambda^{\frac{N}{2}}u_\omega( \lambda x)$ and $S_\omega$ is the action. This result is a complement to the partial result of Ardila and Dinh (Z. Angew. Math. Phys. 2020), where the strong instability of standing waves has been studied under a different assumption.
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Keywords inhomogeneous Gross-Pitaevskii equation; strong instability; ground state

Citation: Yongbin Wang, Binhua Feng. Instability of standing waves for the inhomogeneous Gross-Pitaevskii equation. AIMS Mathematics, 2020, 5(5): 4596-4612. doi: 10.3934/math.2020295

References

  • 1. G. P. Agrawal, Nonlinear Fiber Optics, Academic Press, 2007.
  • 2. G. Baym, C. J. Pethick, Ground state properties of magnetically trapped Bose-Einstein condensate rubidium gas, Phys. Rev. Lett., 76 (1996), 6-9.    
  • 3. L. Pitaevskii, S. Stringari, Bose-Einstein condensation, International Series of Monographs on Physics, 116. The Clarendon Press, Oxford University Press, Oxford, 2003.
  • 4. J. Chen, On a class of nonlinear inhomogeneous Schrödinger equations, J. Appl. Math. Comput., 32 (2010), 237-253.    
  • 5. J. Chen, B. Guo, Sharp global existence and blowing up results for inhomogeneous Schrödinger equations, Discrete Contin. Dyn. Syst. Ser. B, 8 (2007), 357-367.
  • 6. A. de Bouard, R. Fukuizumi, Stability of standing waves for nonlinear Schrödinger equations with inhomogeneous nonlinearities, Ann. Henri Poincaré, 6 (2005), 1157-1177.
  • 7. V. D. Dinh, Blowup of H1 solutions for a class of the focusing inhomogeneous nonlinear Schrödinger equation, Nonlinear Analysis, 174 (2018), 169-188.    
  • 8. B. Feng, On the blow-up solutions for the nonlinear Schrödinger equation with combined powertype nonlinearities, J. Evol. Equ., 18 (2018), 203-220.    
  • 9. B. Feng, H. Zhang, Stability of standing waves for the fractional Schrödinger-Choquard equation, Comput. Math. Appl., 75 (2018), 2499-2507.    
  • 10. F. Genoud, An inhomogeneous, L2-critical, nonlinear Schrödinger equation, Z. Anal. Anwend., 31 (2012), 283-290.    
  • 11. F. Genoud, C. Stuart, Schrödinger equations with a spatially decaying nonlinearity: Existence and stability of standing waves, Discrete Contin. Dyn. Syst., 21 (2008), 137-186.    
  • 12. X. Luo, Stability and multiplicity of standing waves for the inhomogeneous NLS equation with a harmonic potential, Nonlinear Anal. Real World Appl., 45 (2019), 688-703.    
  • 13. J. Zhang, S. Zhu, Sharp energy criteria and singularity of blow-up solutions for the DaveyStewartson system, Commun. Math. Sci., 17 (2019), 653-667.    
  • 14. S. Zhu, Blow-up solutions for the inhomogeneous Schrödinger equation with L2 supercritical nonlinearity, J. Math. Anal. Appl., 409 (2014), 760-776.    
  • 15. T. Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics vol. 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003.
  • 16. S. Le Coz, A note on Berestycki-Cazenave's classical instability result for nonlinear Schrödinger equations, Adv. Nonlinear Stud., 8 (2008), 455-463.
  • 17. A. Bensouilah, V. D. Dinh, S. H. Zhu, On stability and instability of standing waves for the nonlinear Schrödinger equation with inverse-square potential, J. Math. Phys., 59 (2018), 18.
  • 18. J, Chen, B. Guo, Strong instability of standing waves for a nonlocal Schrödinger equation, Physica D: Nonlinear Phenomena, 227 (2007), 142-148.    
  • 19. Z. Cheng, Z. Shen, M. Yang, Instability of standing waves for a generalized Choquard equation with potential, J. Math. Phys., 58 (2017), 13.
  • 20. Z. Cheng, M. Yang, Stability of standing waves for a generalized Choquard equation with potential, Acta Appl. Math., 157 (2018), 25-44.    
  • 21. V. D. Dinh, On instability of standing waves for the mass-supercritical fractional nonlinear Schrödinger equation, Z. Angew. Math. Phys., 70 (2019), 17.
  • 22. B. Feng, Sharp threshold of global existence and instability of standing wave for the SchrödingerHartree equation with a harmonic potential, Nonlinear Anal. Real World Appl., 31 (2016), 132-145.    
  • 23. B. Feng, On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities, Commun. Pure Appl. Anal., 17 (2018), 1785-1804.    
  • 24. B. Feng, R. Chen, Q. Wang, Instability of standing waves for the nonlinear Schrödinger-Poisson equation in the L2-critical case, J. Dynam. Differential Equations, (2019), doi: 10.1007/s10884-019-09779-6.
  • 25. B. Feng, J. Liu, H. Niu, et al. Strong instability of standing waves for a fourth-order nonlinear Schrödinger equation with the mixed dispersions, Nonlinear Anal., 196 (2020), 111791.
  • 26. R. Fukuizumi, M. Ohta, Instability of standing waves for nonlinear Schrödinger equations with potentials, Differ. Integral Equ., 16 (2003), 691-706.
  • 27. R. Fukuizumi, M. Ohta, Strong instability of standing waves with negative energy for double power nonlinear Schrödinger equations, SUT J. Math., 54 (2018), 131-143.
  • 28. M. Ohta, Strong instability of standing waves for nonlinear Schrödinger equations with harmonic potential, Funkcial. Ekvac., 61 (2018), 135-143.    
  • 29. M. Ohta, Strong instability of standing waves for nonlinear Schrödinger equations with a partial confinement, Comm. Pure Appl. Anal., 17 (2018), 1671-1680.    
  • 30. R. Fukuizumi, M. Ohta, Strong instability of standing waves for nonlinear Schrödinger equations with attractive inverse power potential, Osaka J. Math., 56 (2019), 713-726.
  • 31. Y. Wang, Strong instability of standing waves for Hartree equation with harmonic potential, Physica D: Nonlinear Phenomena, 237 (2008), 998-1005.    
  • 32. J. Zhang, Sharp threshold for blowup and global existence in nonlinear Schrödinger equations under a harmonic potential, Comm. Partial Differential Equations, 30 (2005), 1429-1443.    
  • 33. J. Zhang, S. Zhu, Stability of standing waves for the nonlinear fractional Schrödinger equation, J. Dynam. Differential Equations, 29 (2017), 1017-1030.    
  • 34. A. H. Ardila, V. D. Dinh, Some qualitative studies of the focusing inhomogeneous Gross-Pitaevskii equation, Z. Angew. Math. Phys., 71 (2020), 24.
  • 35. L. G. Farah, Global well-posedness and blow-up on the energy space for the inhomogeneous nonlinear Schrödinger equation, J. Evol. Equ., 16 (2016), 193-208.    

 

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