Inhomogeneous Nonlinear Schrdinger Equation with Harmonic Potential

doi:10.3969/j.cnki.jdxb.2016.03.002

Abstract

Abstract:

The inhomogeneous nonlinear Schrödinger equation with harmonic potential is discussed.By establishing the corresponding energy function and the variational structure and through applying the potential well argument,the concave function and the improved Gagliardo-Nirenberg interpolation inequation,the optimum conditions for the global solution and explosive solution for the Cauchy problem are obtained.In addition，the initial value for the existence of global solution is determined.

Key words: inhomogeneous, nonlinear Schrö, dinger equation, global solution, explosive solution, optimum conditions

SHE Zhaobing,HE Xiaofei,CHEN Fengxia. Inhomogeneous Nonlinear Schrdinger Equation with Harmonic Potential[J]. Journal of Jishou University（Natural Sciences Edition), DOI: 10.3969/j.cnki.jdxb.2016.03.002.

References

[1] RABINOWITZ PAUL H.On a Class of Nonlinear Schrdinger Equations[J].Zeitschrift für Angewandte Mathematik und Physik,1992,43(2):270-291.
[2] CAO Daomin,NOUSSAIR EZZAT S.Multi-Bump Standing Waves with Critical Frequency for Nonlinear Schrdinger Equations[J].J. Differential Equations,2004,203:292-312.
[3] ROSE HARVEY A,WEINSTEIN MICHAWLL I.On the Bound States of the Nonlinear Schrdinger Equation with a Linear Potential[J].Phyica D:Nonlinear Phenomena,1988,30(1-2):207-218.
[4] ZHANG Jian.Stability of attractive Bose-Einstein Condensate[J].Journal of Statistical Physics,2002,101(3)：731-746.
[5] ZHANG Jian.Stability of Standing Waves for Nonlinear Schrdinger Equations with Unbounded Potentials[J].Zeitschrift für Angewandte Mathematik und Physik,2000,51(3):498-505.
[6] OH YONG-GEUN.Cauchy Problem and Ehrenfest’s Law of Nonlinear Schrdinger Equations with Potentials[J].Journal of Differential Equations,1989,81(2):255-274.
[7] LEVINE HOWARD A.Instability and Non-Existence of Global Solutions to Nonlinear Wave Equations of the Form Putt=－Au+F(u)[J].Trans. Amer. Math. Soc.,1974,192：1-21.
[8] ZHANG Jian.Sharp Threshold for Blowup and Global Existence in Nonlinear Schrdinger Equations Under a Harmonic Potential[J].Communications in Partial Differential Equations,2005,30(10):1 429-1 443.
[9] CARLES RMI.Remark on Nonlinear Schrdinger Equations with Harmonic Potential[J].Ann. Henri Poincaré,2002,3(4):757-772.
[10] CARLES RMI.Critical Nonlinear Schrdinger Equations with and Without Harmonic Potential[J].Mathematical Models Methods Applied Sciences,2002,12(10):1 513-1 523.
[11] CARLES RMI.Nonlinear Schrdinger Equations with Repulsive Harmonic Potential and Applications[J].SIAM J. Math. Anal.,2003,35(4):823-843.
[12] CARLES RMI.Global Existence Results for Nonlinear Schrdinger Equations with Quadratic Potentials[J].Discrete and Continuous Dynamical Systems,2005,13(2):385-398.
[13] CAZENAVE T.An Introduction to Nonlinear Schrdinger Equations[M]∥Textos de Métodos Mathmátics.Rio de Janeiro:IM-UFRJ,1993.
[14] CHEN Guanggan,ZHANG Jian.Sharp Threshold of Global Existence for Non-Linear Gross-Pitaevskii Equation in R N [J].IMA Journal of Applied Mathematics,2006,71(2):232-240.
[15] CHEN Guanggan,ZHANG Jian.Remarks on Global Existence for the Supercritical Nonlinear Schrdinger Equation with a Harmonic Potential[J].Journal of Mathematical Analysis and Applications,2006,320(2)：591-598.
[16] CHEN Guanggan,ZHANG Jian.A Threshold of Global Existence for the Critical Nonlinear Gross-Pitaevskii Equation[J].Proceedings of the Edinburgh Mathematical Society,2007,50(l):63-71.
[17] BOUARD A DE,DEBUSSCHE A.A Stochastic Nonlinear Schrdinger Equation with Multiplicative Noise[J].Communications in Mathematical Physics,1999,205(1):161-181.
[18] BOUARD A DE,DEBUSSCHE A.On the Effect of a Noise on Solutions of the Focusing Supercritical Nonlinear Schrdinger Equation[J].Probab. Theory Related Fields,2002,123:6-96.
[19] PAYNE L E,SATTINGER D H.Saddle Points and Instability of Nonlinear Hyperbolic Equations[J].Israel Journal Mathematics,1975,22(3):273-303.