具组合型非线性项与调和位势的非线性Schrodinger方程

徐润章; 徐闯

doi:10.3879/j.issn.1000-0887.2010.04.011

具组合型非线性项与调和位势的非线性Schrodinger方程

doi: 10.3879/j.issn.1000-0887.2010.04.011

徐润章¹,
徐闯²

哈尔滨工程大学理学院,哈尔滨 150001;2.哈尔滨工业大学数学系,哈尔滨 150001

基金项目: 国家自然科学基金资助项目(10871055;10926149);黑龙江省自然科学基金资助项目(A200702A200810);黑龙江省教育厅科学技术基金资助项目(11541276);哈尔滨工程大学科学基金

详细信息

作者简介:
徐润章(1982- ),男,河北人,副教授,博士(联系人.E-mail:xurunzh@yahoo.com.cn).

中图分类号: O175.29
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- 被引次数: 0
出版历程
- 收稿日期: 1900-01-01
- 修回日期: 2010-03-12
- 刊出日期: 2010-04-15

Nonlinear Schrdinger Equation With Combined Power-Type Nonlinearities and Harmonic Potential

XU Run-zhang¹,
XU Chuang²

College of Science, Harbin Engineering University, Harbin 150001, P. R. China;

摘要

摘要: 讨论了一类带有组合型非线性项与调和位势的非线性Schrdinger方程．通过构造变分问题，引入位势井方法．给出了位势井的结构和位势井深度函数的性质．得到了问题的相关集合在流之下的不变性．揭示了只要问题的初值属于位势井内或位势井外，则问题在今后所有时间内的解都存在于位势井内或井外．结合凹性方法,解的整体存在性的最佳条件被给出．
- 最佳条件 /
- 不变流形 /
- 调和位势 /
- 组合型非线性项
Abstract: A class of nonlinear Schr^Ldinger equations with combined power-type nonlinearities and harmonic potential are discussed.By constructing a variational problem the potential well method is applied.The structure of the potential well and the properties of depth function are given.The invariance of some sets for the problem is shown.It is proven that if the initial data are in the potential well or out of it,the solutions will lie either in the potential well or out of it respectively.By convexity method,the sharp condition of the global well-posedness is given.
- sharp criterion /
- invariant manifold /
- harmonic potential /
- combined power-type nonlinearities

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参考文献(30)

[1]	ZHANG Jian. Sharp conditions of global existence for nonlinear Schrdinger and Klein-Gordon equations[J]. Nonlinear Analysis， 2002， 48(2):191-207. doi: 10.1016/S0362-546X(00)00180-2
[2]	Ginibre J，Velo G. On a class of nonlinear Schrdinger equations[J].Journal of Functional Analysis， 1979， 32(1):1-71. doi: 10.1016/0022-1236(79)90076-4
[3]	Glassey R T. On the blowing up of solutions to the Cauchy problem for nonlinear Schrdinger equations[J]. Journal of Mathematical Phisics， 1977， 18(9):1794-1797. doi: 10.1063/1.523491
[4]	Ogawa T， Tsutsumi Y. Blow-up of H1 solution for the nonlinear Schrdinger equation[J]. Journal of Differential Equations， 1991， 92(2):317-330. doi: 10.1016/0022-0396(91)90052-B
[5]	Ogawa T， Tsutsumi Y. Blow-up of H1 solution for the nonlinear Schrdinger equation with critical power nonlinearity[J]. Proceedings of the American Mathematical Society， 1991， 111(2):487-496.
[6]	Kenig C， Ponce G， Vega L. Small solution to nonlinear Schrdinger equations[J].Annales de l’Institut Henri Poincaré Nonlinear Analysis， 1993， 10(3):255-288.
[7]	Hayashi N， Nakamitsu K， Tsutsumi M. On solutions of the initial value problem for the nonlinear Schrdinger equations[J].Journal of Functional Analysis， 1987， 71(2):218-245. doi: 10.1016/0022-1236(87)90002-4
[8]	Hayashi N，Tsutsumi Y. Scattering theory for Hartree type equations[J]. Annales de I’institut Henri Poincaré，Physique Théorique， 1987， 46:187-213.
[9]	Ginibre J， Velo G. The global Cauchy problem for the nonlinear Schrdinger equation[J]. Annales de l’Institut Henri Poincaré Non Linear Analysis， 1985， 2(4):309-327.
[10]	Ginibre J， Ozawa T. Long range scattering for nonlinear Schrdinger and Hartree equations in space dimensions n≥2[J]. Communications in Mathematical Physic， 1993， 151(3):619-645. doi: 10.1007/BF02097031
[11]	Strauss W A. Nonlinear Wave Equations[M]. Conference Board of the Mathematical Sciences， No. 73.Providence，Rhode Island:American Mathematical Society，1989.
[12]	Cazenave T. An Introduction to Nonlinear Schrdinger Equations[M]. Textos de Metodos Matematicos， Vol 22，Rio de Janeiro: 1989.
[13]	CHEN Guang-gan， 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. doi: 10.1016/j.jmaa.2005.07.008
[14]	Fujiwara D. Remarks on convergence of the Feynman path integrals[J].Duke Mathematical Journal， 1980， 47(3):559-600. doi: 10.1215/S0012-7094-80-04734-1
[15]	Yajima K. On fundamental solution of time dependent Schrdinger equations[J]. Contemporary Mathematics， 1998， 217:49-68. doi: 10.1090/conm/217/02981
[16]	Oh Y G. Cauchy problem and Ehrenfest’s law of nonlinear Schrdinger equations with potentials[J]. Journal of Differential Equations， 1989， 81(2):255-274. doi: 10.1016/0022-0396(89)90123-X
[17]	ZHANG Jian. Stability of standing waves for nonlinear Schrdinger equations with unbounded potentials[J].Zeitschrift für Angewandte Mathematik und Physik， 2000, 51(3):498-503. doi: 10.1007/PL00001512
[18]	ZHANG Jian. Stability of attractive Bose-Einstein condensates[J].Journal of Statistical Physics， 2000， 101(3/4):731-746. doi: 10.1023/A:1026437923987
[19]	Carles Rémi. Critical nonlinear Schrdinger equation with and without harmonic potential[J]. Mathematical Models and Methods in Applied Sciences， 2002， 12(10):1513-1523. doi: 10.1142/S0218202502002215
[20]	Carles Rémi . Remarks on the nonlinear Schrdinger equation with harmonic potential[J]. Annales Henri Poincaré， 2002， 3(3/4):757-772. doi: 10.1007/s00023-002-8635-4
[21]	Tsurumi T， Wadati M. Collapses of wave functions in multidimensional nonlinear Schrdinger equations under harmonic potential[J]. Journal of the Physical Society of Japan， 1997， 66(10):3031-3034.
[22]	Shu J， Zhang J. Nonlinear Shrdinger equation with harmonic potential[J]. Journal of Mathematical Physics， 2006， 47(6):063503. doi: 10.1063/1.2209168
[23]	Tao Terence，Visan Monica，ZHANG Xiao-yi. The nonlinear Schrdinger equation with combined power-type nonlinearities[J]. Commumications in Partial Differential Equations， 2007， 32(7/9):1281-1343. doi: 10.1080/03605300701588805
[24]	SHU Ji, ZHANG Jian. Instability of standing waves for a class of nonlinear Schrdinger equations[J].Journal of Mathematical Analysis and Applications,2007， 327(2):878-890. doi: 10.1016/j.jmaa.2006.04.082
[25]	XU Run-zhang，ZHANG Wen-ying , WU Wei-ning. Nonlinear Analysis Research Trends[M].Incorporation: Nova Science Publishers，2008: 259-281.
[26]	Payne L E, Sattinger D H. Saddle points and intability of nonlinear hyperbolic equations[J]. Israel Journal of Mathematics， 1975， 22(3/4):273-303. doi: 10.1007/BF02761595
[27]	XU Run-zhang， LIU Ya-cheng. Remarks on nonlinear Schrdinger equation with harmonic potential[J]. Journal of Mathematical Physics， 2008， 49(4):043512. doi: 10.1063/1.2905154
[28]	Kato T. On nonlinear Shrdinger equations[J]. Annales de I’institut Henri Poincaré，Physique Théorique， 1987， 49(1):113-129.
[29]	Cazenave T. Semilinear Schrdinger Equations[M]. Courant Lecture Notes in Mathematics.Providence，Rhode Island: American Mathematical Society，2003.
[30]	Tsutsumi Y, Zhang J. Instability of optical solitons for two-wave interaction model in cubic nonlinear media[J]. Advances in Mathematical Sciences and Applications， 1998， 8(2):691-713.