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 & 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. Google Scholar

[2]

A. R. BishopR. FleshM. G. ForestD. 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.   Google Scholar

[3]

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

[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. Google Scholar

[5]

J. L. BonaS. 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.   Google Scholar

[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.   Google Scholar

[7]

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

[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.   Google Scholar

[9]

D. ChebanJ. 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.   Google Scholar

[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. Google Scholar

[11]

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

[12]

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

[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.   Google Scholar

[14]

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

[15]

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

[16]

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

[17]

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

[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.   Google Scholar

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

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

[21]

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

[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. Google Scholar

[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.   Google Scholar

[24]

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

[25]

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

[26]

W. Strauss, Nonlinear Wave Equations, Providence, RI, (1989).   Google Scholar

[27]

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

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. Google Scholar

[2]

A. R. BishopR. FleshM. G. ForestD. 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.   Google Scholar

[3]

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

[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. Google Scholar

[5]

J. L. BonaS. 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.   Google Scholar

[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.   Google Scholar

[7]

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

[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.   Google Scholar

[9]

D. ChebanJ. 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.   Google Scholar

[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. Google Scholar

[11]

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

[12]

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

[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.   Google Scholar

[14]

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

[15]

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

[16]

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

[17]

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

[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.   Google Scholar

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

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

[21]

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

[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. Google Scholar

[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.   Google Scholar

[24]

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

[25]

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

[26]

W. Strauss, Nonlinear Wave Equations, Providence, RI, (1989).   Google Scholar

[27]

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

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