August  2017, 22(6): 2147-2168. doi: 10.3934/dcdsb.2017089

Averaging principle for the Schrödinger equations

School of Mathematics and Statistics, and Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun 130024, China

Peng Gao, E-mail address: gaopengjilindaxue@126.com

Received  June 2016 Revised  November 2016 Published  March 2017

Fund Project: The first author is supported by NSFC Grant 11601073 and the Fundamental Research Funds for the Central Universities, the second author is supported by NSFC Grant 11171132.

Averaging principle for the cubic nonlinear Schrödinger equations with rapidly oscillating potential and rapidly oscillating force are obtained, both on finite but large time intervals and on the entire time axis. This includes comparison estimate, stability estimate, and convergence result between nonlinear Schrödinger equation and its averaged equation. Furthermore, the existence of almost periodic solution for cubic nonlinear Schrödinger equations is also investigated.

Citation: Peng Gao, Yong Li. Averaging principle for the Schrödinger equations. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2147-2168. doi: 10.3934/dcdsb.2017089
References:
[1]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Nauka, Moscow 1989; English transl. , North-Holland, Amsterdam 1992.

[2]

A. R. Bishop, R. Flesh, M. G. Forest, D. W. McLaughlin and E. A. Overman, Correlations between chaos in a perturbed sine-Gordon equations and a truncated model system, SIAM J. Math. Anal., 21 (1990), 1511-1536.

[3]

N. N. Bogolyubov, On Some Statistical Methods in Mathematical Physics, Izdat. Akad. Nauk Ukr. SSR, Kiev, 1945.

[4]

N. N. Bogoliubov and Y. A. Mitropolsky, Asymptotic Methods in the Theory of Non-Linear Oscillations, Fizmatgiz, Moscow 1963; English transl. , Gordon and Breach, New York, 1962.

[5]

J. L. Bona, S. M. Sun and B. Y. Zhang, A nonhomogeneous boundary-value problem for the Korteweg-de Vries equation posed on a finite domain, Comm. Partial Differential Equations, 28 (2003), 1391-1436.

[6]

J. Bourgain, Fourier transformation restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Part 1: Schrödinger equations, GAFA, 3 (1993), 107-156.

[7]

H. Brezis and T. Gallouet, Nonlinear Schrödinger evolution equations, Nonlinear Analysis: Theory, Methods & Applications, 4 (1980), 677-681.

[8]

D. Cheban and J. Duan, Almost periodic solutions and global attractors of non-autonomous Navier-Stokes equations, Journal of Dynamics and Differential Equations, 16 (2004), 1-34.

[9]

D. Cheban, J. Duan and A. Gherco, Generalization of the second Bogolyubov's theorem for non-almost periodic systems, Nonlinear Analysis: Real World Applications, 4 (2003), 599-613.

[10]

Y. L. Daletskii and M. G. Krein, Stability of solutions of differential equations in Banach space, Nauka, Moscow 1970; English transl. , Araer. Math. Soc, Providence, RI 1974.

[11]

V. P. Dymnikov and A. N. Filatov, Mathematics of Climate Modeling, Birkhaüser, Boston, MA 1997.

[12]

A. N. Filatov, Asymptotic Methods in the Theory of Differential and Integrodifferential Equations, Fan, Tashkent 1974. (Russian)

[13]

H. Gao and J. Duan, Dynamics of quasi-geostrophic fluid motion with rapidly oscillating Coriolis force, Nonlinear Anal. Real World Appl., 4 (2003), 127-138.

[14]

H. Gao and J. Duan, Averaging principle for quasi-geostrophic motion under rapidly oscillating forcing, Applied Mathematics and Mechanics, 26 (2005), 108-120.

[15]

J. M. Ghidaglia, Finite dimensional behavior for weakly damped driven Schrödinger equations, Annales de lIHP Analyse non lineaire, 5 (1998), 365-405.

[16]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, New York 1981.

[17]

A. A. Ilyin, Averaging principle for dissipative dynamical systems with rapidly oscillating right-hand sides, Sbornik: Mathematics, 187 (1996), 635-677.

[18]

A. A. Ilyin, Global averaging of dissipative dynamical system, rendiconti academia nazionale delle scidetta dli XL. Memorie di Matematica e Applicazioni, 22 (1998), 165-191.

[19] B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge Univ. Press, Cambridge, 1982. 
[20]

Yu. A. Mitropolskii, The Method of Averaging in Non-Linear Mechanics, Naukova Dumka, Kiev, 1971. (Russian)

[21]

K. Nozaki and N. Bekky, Low dimensional chaos in a driven damped nonlinear Schrödinger equation, Physica D, 21 (1986), 381-393.

[22]

M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, second ed. , in: Texts in Applied Mathematics, vol. 13, Springer-Verlag, New York, 2004.

[23]

L. Rosier and B. Y. Zhang, Global stabilization of the generalized Korteweg-de Vries equation posed on a finite domain, SIAM Journal on Control and Optimization, 45 (2006), 927-956.

[24]

I. Segal, Non-linear semi-groups, Annals of Mathematics, 78 (1963), 339-364.

[25]

I. B. Simonenko, A justification of the method of averaging for abstract parabolic equations, (Russian) Dokl. Akad. Nauk SSSR, 191 (1970), 33-34.

[26]

W. Strauss, Nonlinear Wave Equations, Providence, RI, 1989.

[27]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Second edition. Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997.

show all references

References:
[1]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Nauka, Moscow 1989; English transl. , North-Holland, Amsterdam 1992.

[2]

A. R. Bishop, R. Flesh, M. G. Forest, D. W. McLaughlin and E. A. Overman, Correlations between chaos in a perturbed sine-Gordon equations and a truncated model system, SIAM J. Math. Anal., 21 (1990), 1511-1536.

[3]

N. N. Bogolyubov, On Some Statistical Methods in Mathematical Physics, Izdat. Akad. Nauk Ukr. SSR, Kiev, 1945.

[4]

N. N. Bogoliubov and Y. A. Mitropolsky, Asymptotic Methods in the Theory of Non-Linear Oscillations, Fizmatgiz, Moscow 1963; English transl. , Gordon and Breach, New York, 1962.

[5]

J. L. Bona, S. M. Sun and B. Y. Zhang, A nonhomogeneous boundary-value problem for the Korteweg-de Vries equation posed on a finite domain, Comm. Partial Differential Equations, 28 (2003), 1391-1436.

[6]

J. Bourgain, Fourier transformation restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Part 1: Schrödinger equations, GAFA, 3 (1993), 107-156.

[7]

H. Brezis and T. Gallouet, Nonlinear Schrödinger evolution equations, Nonlinear Analysis: Theory, Methods & Applications, 4 (1980), 677-681.

[8]

D. Cheban and J. Duan, Almost periodic solutions and global attractors of non-autonomous Navier-Stokes equations, Journal of Dynamics and Differential Equations, 16 (2004), 1-34.

[9]

D. Cheban, J. Duan and A. Gherco, Generalization of the second Bogolyubov's theorem for non-almost periodic systems, Nonlinear Analysis: Real World Applications, 4 (2003), 599-613.

[10]

Y. L. Daletskii and M. G. Krein, Stability of solutions of differential equations in Banach space, Nauka, Moscow 1970; English transl. , Araer. Math. Soc, Providence, RI 1974.

[11]

V. P. Dymnikov and A. N. Filatov, Mathematics of Climate Modeling, Birkhaüser, Boston, MA 1997.

[12]

A. N. Filatov, Asymptotic Methods in the Theory of Differential and Integrodifferential Equations, Fan, Tashkent 1974. (Russian)

[13]

H. Gao and J. Duan, Dynamics of quasi-geostrophic fluid motion with rapidly oscillating Coriolis force, Nonlinear Anal. Real World Appl., 4 (2003), 127-138.

[14]

H. Gao and J. Duan, Averaging principle for quasi-geostrophic motion under rapidly oscillating forcing, Applied Mathematics and Mechanics, 26 (2005), 108-120.

[15]

J. M. Ghidaglia, Finite dimensional behavior for weakly damped driven Schrödinger equations, Annales de lIHP Analyse non lineaire, 5 (1998), 365-405.

[16]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, New York 1981.

[17]

A. A. Ilyin, Averaging principle for dissipative dynamical systems with rapidly oscillating right-hand sides, Sbornik: Mathematics, 187 (1996), 635-677.

[18]

A. A. Ilyin, Global averaging of dissipative dynamical system, rendiconti academia nazionale delle scidetta dli XL. Memorie di Matematica e Applicazioni, 22 (1998), 165-191.

[19] B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge Univ. Press, Cambridge, 1982. 
[20]

Yu. A. Mitropolskii, The Method of Averaging in Non-Linear Mechanics, Naukova Dumka, Kiev, 1971. (Russian)

[21]

K. Nozaki and N. Bekky, Low dimensional chaos in a driven damped nonlinear Schrödinger equation, Physica D, 21 (1986), 381-393.

[22]

M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, second ed. , in: Texts in Applied Mathematics, vol. 13, Springer-Verlag, New York, 2004.

[23]

L. Rosier and B. Y. Zhang, Global stabilization of the generalized Korteweg-de Vries equation posed on a finite domain, SIAM Journal on Control and Optimization, 45 (2006), 927-956.

[24]

I. Segal, Non-linear semi-groups, Annals of Mathematics, 78 (1963), 339-364.

[25]

I. B. Simonenko, A justification of the method of averaging for abstract parabolic equations, (Russian) Dokl. Akad. Nauk SSSR, 191 (1970), 33-34.

[26]

W. Strauss, Nonlinear Wave Equations, Providence, RI, 1989.

[27]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Second edition. Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997.

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