Blow-Up and Attractor of Solution for Problems of Nonlinear Schrodinger Equations

ABSTRACT

In this paper, the authors study the blow-up of solution for a class of nonlinear Schrodinger equation for some initial boundary problem. On the other hand, the authors give out some analyses and that new conclusion by Eigen-function method. In last section, the authors check the nonlinear parameter for light rule power by using of parameter method to get ground state and excite state correspond case, and discuss the global attractor of some fraction order case, and combine numerical test. To illustrate this physics meaning in dimension*d *= 1, 2 case. So, by numerable solution to give out these wave expression.

In this paper, the authors study the blow-up of solution for a class of nonlinear Schrodinger equation for some initial boundary problem. On the other hand, the authors give out some analyses and that new conclusion by Eigen-function method. In last section, the authors check the nonlinear parameter for light rule power by using of parameter method to get ground state and excite state correspond case, and discuss the global attractor of some fraction order case, and combine numerical test. To illustrate this physics meaning in dimension

KEYWORDS

Nonlinear Schrodinger Equation; Eigen-Function Method; Fractional Order; Blow-Up; Glabal Attractor

Nonlinear Schrodinger Equation; Eigen-Function Method; Fractional Order; Blow-Up; Glabal Attractor

Cite this paper

N. Chen and J. Chen, "Blow-Up and Attractor of Solution for Problems of Nonlinear Schrodinger Equations,"*Applied Mathematics*, Vol. 3 No. 12, 2012, pp. 1921-1932. doi: 10.4236/am.2012.312263.

N. Chen and J. Chen, "Blow-Up and Attractor of Solution for Problems of Nonlinear Schrodinger Equations,"

References

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[3] G. G. Cheng and J. Zhang, “Remark on Global Existence for the Superetitical Nonlinear Shrodinger Equation with a Harmonic Potential,” Journal of Mathematical Analysis and Applications, Vol. 320, No. 2, 2006, pp. 591-598. doi:10.1016/j.jmaa.2005.07.008

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[5] J. S. Zhao, Q. D. Guo, H. O. Yang and R. Z. Xu, “Blow-Up of Solutions for Initial Value Boundary Problem of a Class of Generalized Non-Linear Schrodinger Equation,” Journal of Nature of Science of Heilongjiang University, Vol. 25, No. 2, 2008, pp. 170-172.

[6] N. H. Sweilam and R. F. AI-Bar, “Variational Iteration Method for Coupled Nonlinear Schrodinger Equations,” Computers and Mathematics with Applications, Vol. 54, No. 7-8, 2007, pp. 993-999. doi:10.1016/j.camwa.2006.12.068

[7] J. Li and J. Zhan, “Blow-Up for the Stochastic Nonlinear Schrodinger Equation with a Harmonic Potential,” Advances in Mathematics, Vol. 39, No. 4, 2010, pp. 491-499.

[8] J. Zhang, “Sharp Conditions of Global Existence for Nonlinear Schrodinger and Klein-Golden Equations,” Nonlinear Analysis, Vol. 48, No. 1, 2002, pp. 191-207. doi:10.1016/S0362-546X(00)00180-2

[9] F. Genoud and C. A. Stuart, “Schrodinger Equations with a Spatially Decaying Non-Linear Existence and Stability of Standing Waves,” Discrete and Continuous Dynamical Systems, Vol. 21, No. 1, 2008, pp. 137-186.

[10] S. L. Xu, J. C. Liang and L. Yi, “Exact Solution to a Generalized Nonlinear Schrodinger Equation,” Communications in Theoretical Physics, Vol. 53, No. 1, 2010, pp. 159-165.

[11] H. Zhu, Y. Han and J. Zhang, “Blow-Up of Rough Solutions to the Fourth-Order Nonlinear Schrodinger Equation,” Nonlinear Analysts, Vol. 74, No. 17, 2011, pp. 6186-6201. doi:10.1016/j.na.2011.05.096

[12] H. Meng, B. Tian, T. Xu and H. Q. Zhang, “Backland Transformation and Conservation Laws for the Variable-Coefficient N-Coupled Schrodinger Equations with Symbolic Computation,” Acta Mathematica Sinica, Vol. 28, No. 5, 2012, pp. 969-974. doi:10.1007/s10114-011-0531-8

[13] G. R. Jia, J. C. Zhang, X. Z. Hang and Z. Z. Ren, “Coherent Control of Population Transfer in Li Atoms via Chirped Microwave Pulses,” Chinese Physics Letters, Vol. 26, No. 10, 2009, Article ID: 103201-1-4.

[14] N. H. Sweilam and R. F. Ai-Bar, “Variational Iterative Method for Coupled Nonlinear Schrodinger Equations,” Computers and Mathematics with Applications, Vol. 54, No. 7-8, 2007, pp. 993-999. doi:10.1016/j.camwa.2006.12.068

[15] C. S. Zhu, “An Estimate of the Global Attractor for the Non-Linear Schrodinger Equation with Harmonic Potential,” Journal of Southwest Normal university, Vol. 30, No. 5, 2005, pp. 788-791.

[16] B. L. Guo, T. Q. Han and X. N. Jie, “Existence of Global Smooth Solution to the Periodic Boundary of Fractional Non-Linear Schrodinger Equation,” Applied Mathematics and Computation, Vol. 204, No. 1, 2008, pp. 468-477. doi:10.1016/j.amc.2008.07.003

[17] G. G. Lin and H. J. Gao, “Asymptotic Dynamical Difference between the Nonlocal and Local Swift-Hohenberg Models,” Journal Mathematical Physics, Vol. 41, No. 4, 2000, pp. 2077-2089. doi:10.1063/1.533228

[18] L. Wang. J. B. Dang and G. G. Ling, “The Global Attractor of the Fractional Nonlinear Schrodinger Equation and the Estimate of Its Dimension,” Journal of Yunnan University, Vol. 32, No. 2, 2010, pp. 130-135.

[19] V. G. Makhankov, “On Stationary Solutions of Schrodinger Equation with a Self-Consistent Satisfying Boussinesq’s Equations,” Physics Letters A, Vol. 500, 1974, pp. 42-44.

[20] S. Zhang and F. Wang, “Inter Effects between There Coupled Bose-Einstein Condensates,” Physics Letters A, Vol. 279, 2001, pp. 231-238.

[21] A. Aftalion and Q. Do, “Vortices in A Rotating Bose-Einstein Condensate: Critical Angular Velocities and Energy Diagrams in the Thomas-Fermi regime,” Physics Letters A, Vol. 64, No. 6, 2001, Article ID: 063603. doi:10.1103/PhysRevA.64.063603

[22] C. Tozzo, M. Kramer and F. Dalfovo, “Stability Diagram and Growth Rate of Parametric Resonances in Bose-Einstein Condensates in One Dimensional Optical Lattices,” Physics Letters A, Vol. 72, No. 2, 2005, Article ID: 023613. doi:10.1103/PhysRevA.72.023613

[23] J. Y. Zeng, “Quantum Mechanics,” Science Press, Beijing, 2004, pp. 33-144.

[24] R. Teman, “Infinite Dimensional Dynamical Systems in Mechanics and Physics,” Springer Verlag, New York, 1988. doi:10.1007/978-1-4684-0313-8

[1] B. L. Guo, “Initial Boundary Value Problem for One Class of System of Multi-Dimension Inhomogeneous GBBM Equation,” Chinese Annals of Mathematics, Vol. 8, No. 2, 1987, pp. 226-238.

[2] B. L. Guo and C. X. Miao, “On Inhomogeneous GBBM Equation,” Journal of Partial Differential Equations, Vol. 8, No. 3, 1995, pp. 193-204.

[3] G. G. Cheng and J. Zhang, “Remark on Global Existence for the Superetitical Nonlinear Shrodinger Equation with a Harmonic Potential,” Journal of Mathematical Analysis and Applications, Vol. 320, No. 2, 2006, pp. 591-598. doi:10.1016/j.jmaa.2005.07.008

[4] J. Zhang, “Blow-Up of Solutions to the Mixed Problems for Nonlinear Schrodinger Equations,” Journal of Sichuan Normal University, Vol. 3, 1989, pp. 1-8.

[5] J. S. Zhao, Q. D. Guo, H. O. Yang and R. Z. Xu, “Blow-Up of Solutions for Initial Value Boundary Problem of a Class of Generalized Non-Linear Schrodinger Equation,” Journal of Nature of Science of Heilongjiang University, Vol. 25, No. 2, 2008, pp. 170-172.

[6] N. H. Sweilam and R. F. AI-Bar, “Variational Iteration Method for Coupled Nonlinear Schrodinger Equations,” Computers and Mathematics with Applications, Vol. 54, No. 7-8, 2007, pp. 993-999. doi:10.1016/j.camwa.2006.12.068

[7] J. Li and J. Zhan, “Blow-Up for the Stochastic Nonlinear Schrodinger Equation with a Harmonic Potential,” Advances in Mathematics, Vol. 39, No. 4, 2010, pp. 491-499.

[8] J. Zhang, “Sharp Conditions of Global Existence for Nonlinear Schrodinger and Klein-Golden Equations,” Nonlinear Analysis, Vol. 48, No. 1, 2002, pp. 191-207. doi:10.1016/S0362-546X(00)00180-2

[9] F. Genoud and C. A. Stuart, “Schrodinger Equations with a Spatially Decaying Non-Linear Existence and Stability of Standing Waves,” Discrete and Continuous Dynamical Systems, Vol. 21, No. 1, 2008, pp. 137-186.

[10] S. L. Xu, J. C. Liang and L. Yi, “Exact Solution to a Generalized Nonlinear Schrodinger Equation,” Communications in Theoretical Physics, Vol. 53, No. 1, 2010, pp. 159-165.

[11] H. Zhu, Y. Han and J. Zhang, “Blow-Up of Rough Solutions to the Fourth-Order Nonlinear Schrodinger Equation,” Nonlinear Analysts, Vol. 74, No. 17, 2011, pp. 6186-6201. doi:10.1016/j.na.2011.05.096

[12] H. Meng, B. Tian, T. Xu and H. Q. Zhang, “Backland Transformation and Conservation Laws for the Variable-Coefficient N-Coupled Schrodinger Equations with Symbolic Computation,” Acta Mathematica Sinica, Vol. 28, No. 5, 2012, pp. 969-974. doi:10.1007/s10114-011-0531-8

[13] G. R. Jia, J. C. Zhang, X. Z. Hang and Z. Z. Ren, “Coherent Control of Population Transfer in Li Atoms via Chirped Microwave Pulses,” Chinese Physics Letters, Vol. 26, No. 10, 2009, Article ID: 103201-1-4.

[14] N. H. Sweilam and R. F. Ai-Bar, “Variational Iterative Method for Coupled Nonlinear Schrodinger Equations,” Computers and Mathematics with Applications, Vol. 54, No. 7-8, 2007, pp. 993-999. doi:10.1016/j.camwa.2006.12.068

[15] C. S. Zhu, “An Estimate of the Global Attractor for the Non-Linear Schrodinger Equation with Harmonic Potential,” Journal of Southwest Normal university, Vol. 30, No. 5, 2005, pp. 788-791.

[16] B. L. Guo, T. Q. Han and X. N. Jie, “Existence of Global Smooth Solution to the Periodic Boundary of Fractional Non-Linear Schrodinger Equation,” Applied Mathematics and Computation, Vol. 204, No. 1, 2008, pp. 468-477. doi:10.1016/j.amc.2008.07.003

[17] G. G. Lin and H. J. Gao, “Asymptotic Dynamical Difference between the Nonlocal and Local Swift-Hohenberg Models,” Journal Mathematical Physics, Vol. 41, No. 4, 2000, pp. 2077-2089. doi:10.1063/1.533228

[18] L. Wang. J. B. Dang and G. G. Ling, “The Global Attractor of the Fractional Nonlinear Schrodinger Equation and the Estimate of Its Dimension,” Journal of Yunnan University, Vol. 32, No. 2, 2010, pp. 130-135.

[19] V. G. Makhankov, “On Stationary Solutions of Schrodinger Equation with a Self-Consistent Satisfying Boussinesq’s Equations,” Physics Letters A, Vol. 500, 1974, pp. 42-44.

[20] S. Zhang and F. Wang, “Inter Effects between There Coupled Bose-Einstein Condensates,” Physics Letters A, Vol. 279, 2001, pp. 231-238.

[21] A. Aftalion and Q. Do, “Vortices in A Rotating Bose-Einstein Condensate: Critical Angular Velocities and Energy Diagrams in the Thomas-Fermi regime,” Physics Letters A, Vol. 64, No. 6, 2001, Article ID: 063603. doi:10.1103/PhysRevA.64.063603

[22] C. Tozzo, M. Kramer and F. Dalfovo, “Stability Diagram and Growth Rate of Parametric Resonances in Bose-Einstein Condensates in One Dimensional Optical Lattices,” Physics Letters A, Vol. 72, No. 2, 2005, Article ID: 023613. doi:10.1103/PhysRevA.72.023613

[23] J. Y. Zeng, “Quantum Mechanics,” Science Press, Beijing, 2004, pp. 33-144.

[24] R. Teman, “Infinite Dimensional Dynamical Systems in Mechanics and Physics,” Springer Verlag, New York, 1988. doi:10.1007/978-1-4684-0313-8