Research article

Instability of standing waves for the inhomogeneous Gross-Pitaevskii equation

  • Received: 22 February 2020 Accepted: 08 May 2020 Published: 22 May 2020
  • MSC : 35Q55, 35A15

  • 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 $\omega_* \gt 0$ such that for all $\omega \gt \omega_*$, 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.

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

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  • 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 $\omega_* \gt 0$ such that for all $\omega \gt \omega_*$, 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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    [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. doi: 10.1103/PhysRevLett.76.6
    [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. doi: 10.1007/s12190-009-0246-5
    [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. doi: 10.1016/j.na.2018.04.024
    [8] B. Feng, On the blow-up solutions for the nonlinear Schrödinger equation with combined powertype nonlinearities, J. Evol. Equ., 18 (2018), 203-220. doi: 10.1007/s00028-017-0397-z
    [9] B. Feng, H. Zhang, Stability of standing waves for the fractional Schrödinger-Choquard equation, Comput. Math. Appl., 75 (2018), 2499-2507. doi: 10.1016/j.camwa.2017.12.025
    [10] F. Genoud, An inhomogeneous, L2-critical, nonlinear Schrödinger equation, Z. Anal. Anwend., 31 (2012), 283-290. doi: 10.4171/ZAA/1460
    [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. doi: 10.3934/dcds.2008.21.137
    [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. doi: 10.1016/j.nonrwa.2018.07.031
    [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. doi: 10.4310/CMS.2019.v17.n3.a4
    [14] S. Zhu, Blow-up solutions for the inhomogeneous Schrödinger equation with L2 supercritical nonlinearity, J. Math. Anal. Appl., 409 (2014), 760-776. doi: 10.1016/j.jmaa.2013.07.029
    [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. doi: 10.1016/j.physd.2007.01.004
    [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. doi: 10.1007/s10440-018-0162-5
    [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. doi: 10.1016/j.nonrwa.2016.01.012
    [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. doi: 10.3934/cpaa.2018085
    [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. doi: 10.1619/fesi.61.135
    [29] M. Ohta, Strong instability of standing waves for nonlinear Schrödinger equations with a partial confinement, Comm. Pure Appl. Anal., 17 (2018), 1671-1680. doi: 10.3934/cpaa.2018080
    [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. doi: 10.1016/j.physd.2007.11.018
    [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. doi: 10.1080/03605300500299539
    [33] J. Zhang, S. Zhu, Stability of standing waves for the nonlinear fractional Schrödinger equation, J. Dynam. Differential Equations, 29 (2017), 1017-1030. doi: 10.1007/s10884-015-9477-3
    [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. doi: 10.1007/s00028-015-0298-y
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