On the inhomogeneous nonlinear Schrödinger equation with harmonic potential and unbounded coefficient

Jianqing Chen

Czechoslovak Mathematical Journal (2010)

  • Volume: 60, Issue: 3, page 715-736
  • ISSN: 0011-4642

Abstract

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By deriving a variant of interpolation inequality, we obtain a sharp criterion for global existence and blow-up of solutions to the inhomogeneous nonlinear Schrödinger equation with harmonic potential i ϕ t = - ϕ + | x | 2 ϕ - | x | b | ϕ | p - 2 ϕ . We also prove the existence of unstable standing-wave solutions via blow-up under certain conditions on the unbounded inhomogeneity and the power of nonlinearity, as well as the frequency of the wave.

How to cite

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Chen, Jianqing. "On the inhomogeneous nonlinear Schrödinger equation with harmonic potential and unbounded coefficient." Czechoslovak Mathematical Journal 60.3 (2010): 715-736. <http://eudml.org/doc/38038>.

@article{Chen2010,
abstract = {By deriving a variant of interpolation inequality, we obtain a sharp criterion for global existence and blow-up of solutions to the inhomogeneous nonlinear Schrödinger equation with harmonic potential \[ \{\rm i\}\varphi \_t=-\triangle \varphi +|x|^2\varphi -|x|^b|\varphi |^\{p-2\}\varphi . \] We also prove the existence of unstable standing-wave solutions via blow-up under certain conditions on the unbounded inhomogeneity and the power of nonlinearity, as well as the frequency of the wave.},
author = {Chen, Jianqing},
journal = {Czechoslovak Mathematical Journal},
keywords = {interpolation inequality; inhomogeneous nonlinear Schrödinger equation; harmonic potential; blow-up; global existence; standing waves; strong instability; interpolation inequality; inhomogeneous nonlinear Schrödinger equation; harmonic potential; blow-up; global existence; standing wave; strong instability},
language = {eng},
number = {3},
pages = {715-736},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the inhomogeneous nonlinear Schrödinger equation with harmonic potential and unbounded coefficient},
url = {http://eudml.org/doc/38038},
volume = {60},
year = {2010},
}

TY - JOUR
AU - Chen, Jianqing
TI - On the inhomogeneous nonlinear Schrödinger equation with harmonic potential and unbounded coefficient
JO - Czechoslovak Mathematical Journal
PY - 2010
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 60
IS - 3
SP - 715
EP - 736
AB - By deriving a variant of interpolation inequality, we obtain a sharp criterion for global existence and blow-up of solutions to the inhomogeneous nonlinear Schrödinger equation with harmonic potential \[ {\rm i}\varphi _t=-\triangle \varphi +|x|^2\varphi -|x|^b|\varphi |^{p-2}\varphi . \] We also prove the existence of unstable standing-wave solutions via blow-up under certain conditions on the unbounded inhomogeneity and the power of nonlinearity, as well as the frequency of the wave.
LA - eng
KW - interpolation inequality; inhomogeneous nonlinear Schrödinger equation; harmonic potential; blow-up; global existence; standing waves; strong instability; interpolation inequality; inhomogeneous nonlinear Schrödinger equation; harmonic potential; blow-up; global existence; standing wave; strong instability
UR - http://eudml.org/doc/38038
ER -

References

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  1. Baym, G., Pethick, C. J., 10.1103/PhysRevLett.76.6, Phys. Rev. Lett. 76 (1996), 6-9. (1996) DOI10.1103/PhysRevLett.76.6
  2. Benjamin, T. B., The stability of solitary waves, Proc. Royal Soc. London, Ser. A. 328 (1972), 153-183. (1972) MR0338584
  3. Berestycki, H., Cazenave, T., Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéarires, C. R. Acad. Sci. Paris I 293 (1981), 489-492. (1981) MR0646873
  4. Bona, J. L., On the stability theory of solitary waves, Proc. Royal Soc. London, Ser. A. 344 (1975), 363-374. (1975) Zbl0328.76016MR0386438
  5. Caffarelli, L., Kohn, R., Nirenberg, L., First order interpolation inequalities with weights, Compositio Math. 53 (1984), 259-275. (1984) Zbl0563.46024MR0768824
  6. Cazenave, T., An Introduction to Nonlinear Schrödinger Equations, Textos de Metodos Matematicos, 22, Rio de Janeiro (1989). (1989) 
  7. Cazenave, T., Lions, P. L., 10.1007/BF01403504, Comm. Math. Phys. 85 (1982), 549-561. (1982) MR0677997DOI10.1007/BF01403504
  8. Chen, J., Guo, B., 10.1016/j.physd.2007.01.004, Phys. D 227 (2007), 142-148. (2007) Zbl1116.35111MR2332502DOI10.1016/j.physd.2007.01.004
  9. Chen, J., Guo, B., 10.3934/dcdsb.2007.8.357, Discrete Contin. Dynam. Systems 8 (2007), 357-367. (2007) Zbl1151.35089MR2317813DOI10.3934/dcdsb.2007.8.357
  10. Fibich, G., Wang, X. P., 10.1016/S0167-2789(02)00626-7, Phys. D. 175 (2003), 96-108. (2003) MR1957907DOI10.1016/S0167-2789(02)00626-7
  11. Fukuizumi, R., 10.3934/dcds.2001.7.525, Discrete Contin. Dyn. Syst. 7 (2001), 525-544. (2001) Zbl0992.35094MR1815766DOI10.3934/dcds.2001.7.525
  12. Fukuizumi, R., Ohta, M., Stability of standing waves for nonlinear Schrödinger equations with potentials, Differential Integral Equations 16 (2003), 111-128. (2003) Zbl1031.35132MR1948875
  13. Fukuizumi, R., Ohta, M., 10.1215/kjm/1250282971, J. Math. Kyoto Univ. 45 (2005), 145-158. (2005) MR2138804DOI10.1215/kjm/1250282971
  14. Gill, T. S., Optical guiding of laser beam in nonuniform plasma, Pramana Journal of Physics 55 (2000), 845-852. (2000) 
  15. Ginibre, J., Velo, G., 10.1016/0022-1236(79)90076-4, J. Funct. Anal. 32 (1979), 1-32, 33-71. (1979) MR0533219DOI10.1016/0022-1236(79)90076-4
  16. Glassey, R. T., 10.1063/1.523491, J. Math. Phys. 18 (1977), 1794-1797. (1977) MR0460850DOI10.1063/1.523491
  17. Grillakis, M., Shatah, J., Strauss, W., 10.1016/0022-1236(87)90044-9, J. Funct. Anal. 74 (1987), 160-197. (1987) Zbl0656.35122MR0901236DOI10.1016/0022-1236(87)90044-9
  18. Liu, C. S., Tripathi, V. K., 10.1063/1.870501, Phys. Plasmas 1 (1994), 3100-3103. (1994) DOI10.1063/1.870501
  19. Liu, Y., Wang, X. P., Wang, K., 10.1090/S0002-9947-05-03763-3, Trans. Amer. Math. Soc. 358 (2006), 2105-2122. (2006) MR2197450DOI10.1090/S0002-9947-05-03763-3
  20. Merle, F., Nonexistence of minimal blow up solutions of equations i u t = - u - K ( x ) | u | 4 / N u in N , Ann. Inst. H. Poincaré, Phys. Théor. 64 (1996), 33-85. (1996) Zbl0846.35060MR1378233
  21. Yong-Geun, Oh, 10.1016/0022-0396(89)90123-X, J. Differential Equations 81 (1989), 255-274. (1989) MR1016082DOI10.1016/0022-0396(89)90123-X
  22. Rose, H. A., Weinstein, M. I., 10.1016/0167-2789(88)90107-8, Phys. D 30 (1988), 207-218. (1988) MR0939275DOI10.1016/0167-2789(88)90107-8
  23. Rother, W., 10.1080/03605309908820733, Comm. Partial Differential Equations 15 (1990), 1461-1473. (1990) MR1077474DOI10.1080/03605309908820733
  24. Shatah, J., Strauss, W., 10.1007/BF01212446, Comm. Math. Phys. 100 (1985), 173-190. (1985) Zbl0603.35007MR0804458DOI10.1007/BF01212446
  25. Sintzoff, P., Willem, M., A semilinear elliptic equation on N with unbounded coefficients, Variational and topological methods in the study of nonlinear phenomena 49 (Pisa 2000) 105-113 Birkhauser, Boston, 2002. MR1879738
  26. Strauss, W., 10.1007/BF01626517, Comm. Math. Phys. 55 (1977), 149-162. (1977) Zbl0356.35028MR0454365DOI10.1007/BF01626517
  27. Tsurumi, T., Waditi, M., 10.1143/JPSJ.66.3031, J. Phys. Soc. Japan 66 (1997), 3031-3034. (1997) DOI10.1143/JPSJ.66.3031
  28. Tsurumi, T., Waditi, M., Instability of the Bose-Einstein condensate under magnetic trap, J. Phys. Soc. Japan 66 (1997), 3035-3039. (1997) 
  29. Wang, Y., 10.1016/j.physd.2007.11.018, Phys. D 237 (2008), 998-1005. (2008) Zbl1143.35372MR2417084DOI10.1016/j.physd.2007.11.018
  30. Weinstein, M. I., 10.1007/BF01208265, Comm. Math. Phys. 87 (1983), 567-576. (1983) Zbl0527.35023MR0691044DOI10.1007/BF01208265
  31. Willem, M., Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications, 24, Birkhäuser, Boston (1996). (1996) Zbl0856.49001MR1400007
  32. Zhang, J., 10.1080/03605300500299539, Comm. Partial Differential Equations 30 (2005), 1429-1443. (2005) MR2182299DOI10.1080/03605300500299539

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