On a Class of Quasilinear Schr<sup>L</sup>dinger Equations

SHU Ji; ZHANG Jian

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2007 > 28(7): 877-882

SHU Ji, ZHANG Jian. On a Class of Quasilinear SchrLdinger Equations[J]. Applied Mathematics and Mechanics, 2007, 28(7): 877-882.

Citation:

SHU Ji, ZHANG Jian. On a Class of Quasilinear Schr^Ldinger Equations[J]. Applied Mathematics and Mechanics, 2007, 28(7): 877-882.

SHU Ji, ZHANG Jian. On a Class of Quasilinear SchrLdinger Equations[J]. Applied Mathematics and Mechanics, 2007, 28(7): 877-882.

Citation:

SHU Ji, ZHANG Jian. On a Class of Quasilinear Schr^Ldinger Equations[J]. Applied Mathematics and Mechanics, 2007, 28(7): 877-882.

PDF( 261 KB)

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;

Received Date: 2006-09-04
Rev Recd Date: 2007-04-27
Publish Date: 2007-07-15

Abstract

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

FullText(HTML)

References(22)

References

[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.