一类拟线性Schrödinger方程

舒级; 张健

一类拟线性Schrödinger方程

舒级^1,2,
张健²

1.
四川师范大学计算机软件实验室,成都 610066;2.四川师范大学数学与软件科学学院, 成都 610066

基金项目: 四川省教育厅资助科研项目

详细信息

作者简介:
舒级(1977- ),男,四川纳溪人,讲师,博士(联系人.Tel:+86-28-84761502;E-mail:shu.ji@163.com;shuji2008@hotmail.com).

中图分类号: O175.27
计量
- 文章访问数: 3188
- HTML全文浏览量: 194
- PDF下载量: 636
- 被引次数: 0
出版历程
- 收稿日期: 2006-09-04
- 修回日期: 2007-04-27
- 刊出日期: 2007-07-15

On a Class of Quasilinear Schr^Ldinger Equations

SHU Ji^1,2,
ZHANG Jian²

1.
Sichuan Provincial Key Laboratory of Computer Software, Sichuan Normal University, Chengdu 610066, P. R. China;

摘要

摘要: 在二维空间中讨论一类拟线性Schrdinger方程,该方程在物理学上描述了吸引玻色-爱因斯坦凝聚．通过建立这个方程的性质,运用能量方法,证明了该方程所对应的初值问题的解在一定条件下爆破．同时利用变分方法，也得到了整体解存在的一个充分条件,该条件与一个经典的椭圆方程的基态有关．
- 拟线性Schrdinger方程 /
- 爆破 /
- 整体解 /
- 基态 /
- 玻色-爱因斯坦凝聚
Abstract: A type of quasilinear Schr^Ldinger equations in two dimensions are discussed, which describe attractive Bose-Einstein Condensates in physics. By establishing the property of the equation and applying the energymethod, was proved the blowup of the solutions to the Cauchy problem for the equation under certain conditions. At the same time, by the variational method, the a sufficient condition of global existence was got, which is related to the ground state of a classical elliptic equation.
- quasilinear Schr^Ldinger equations /
- blow up /
- global solution /
- ground state /
- Bose-Einstein condensates

HTML全文

参考文献(22)

[1]	Laedke E W,Spatschek K H,Stenflo L.Evolution theorem for a class of perturbed envelope soliton solutions[J].J Math Phys，1983,24(12):2764-2769. doi: 10.1063/1.525675
[2]	De Bouard A,Hayashi N,Saut J C.Global existence of small solutions to a relativistic nonlinear Schrdinger equation[J].Comm Math Phys,1997,189(1):73-105. doi: 10.1007/s002200050191
[3]	Nakamura A.Damping and modification of exciton solitary waves[J].J Phys Soc Japan,1977,42(6):1824-1835. doi: 10.1143/JPSJ.42.1824
[4]	Porkolab M,Goddman M V. Upper-hybrid solitons and oscillating two-stream instabilities[J].Phys Fluids,1976,19(6):872-881. doi: 10.1063/1.861553
[5]	Quispel G R W,Capel H W. Equation of motion for the Heisenberg spin chain[J].Physica A,1982,110(1/2):41-80. doi: 10.1016/0378-4371(82)90104-2
[6]	Takeno S, Homma S. Classical planar Heisenberg ferromagnet,complex scalar fields and nonlinear excitations[J].Progr Theoret Physics,1981,65(1):172-189. doi: 10.1143/PTP.65.172
[7]	Hasse R W. A general method for the solution of nonlinear soliton and kink Schrdinger equations[J].Z Physik B,1980,37(1):83-87. doi: 10.1007/BF01325508
[8]	Makhankov V G,Fedyanin V K. Non-linear effects in quasi-one-dimensional models of condensed matter theory[J].Physics Reports,1984,104(1):1-86. doi: 10.1016/0370-1573(84)90106-6
[9]	Poppenberg M. Smooth solutions for a class of fully nonlinear Schrdinger type equations[J].Nonlinear Analysis TMA,2001,45(6):723-741. doi: 10.1016/S0362-546X(99)00436-8
[10]	Poppenberg M.On the local well posedness of quasi-linear Schrdinger equations in arbitrary space dimension[J].J Differnatial Equations,2001,172(1):83-115. doi: 10.1006/jdeq.2000.3853
[11]	Poppenberg M. An inverse function theorem in Sobolev spaces and applications to quasi-linear Schrdinger equations[J].J Math Anal Appl,2001,258(1):146-170. doi: 10.1006/jmaa.2000.7366
[12]	Poppenberg M,Schmitt K, Wang Z Q. On the existence of soliton solutions to quasi-linear Schrdinger equations[J].Calc var Partial Differential Equations,2002,14(3):329-344. doi: 10.1007/s005260100105
[13]	Liu J Q, Wang Y Q, Wang Z Q. Soliton solutions for quasi-linear Schrdinger equations Ⅱ[J].J Differnatial Equations,2003,187(2):473-493. doi: 10.1016/S0022-0396(02)00064-5
[14]	Juan J. García,Vladimir V. Konotop, Boris Malomed. A quasi-local Gross-Pitaevskii equation for attractive Bose-Einstein condensates[J].Mathematics and Computers in Simulation,2003,62(1/2):21-30. doi: 10.1016/S0378-4754(02)00190-8
[15]	张健.具非线性二阶导数项的Schrdinger方程混合问题的爆破性质[J].数学物理学报,1994,14(增刊):89-94.
[16]	Glassey R T.On the blowup of nonlinear Schrdinger equations[J].J Math Phys,1977,18(9):1794-1797. doi: 10.1063/1.523491
[17]	Tsutsumi Y,Zhang Jian.Instability of optical solitions for two-wave interaction model in cubic nonlinear media[J].Adv Math Sci Appl,1998,8(2):691-713.
[18]	Weinstein M I.Nonlinear Schrdinger equations and sharp interpolations estimates[J].Comm Math Phys,1983,87(4):567-576. doi: 10.1007/BF01208265
[19]	Kwong M K. Uniqueness of positive solutions of Δu-u+up=0 in RN[J].Arch Ration Mech Anal,1989,105(3):243-266.
[20]	ZHANG Jian. Stability of attractive Bose-Einstein condensates[J].J Stat Phys,2000,101(3/4):731-746. doi: 10.1023/A:1026437923987
[21]	Kavian O. A remark on the blowing up of solutions to the Cauchy problem for nonlinear Schrdinger equations[J].Trans Amer Math Soc,1987,299(1):193-203.
[22]	Cazenave T. An Introduction to Nonlinear Schrdinger Equations[M].Rio de Janeiro:Textos de Metods Matematicos, 1989.