September  2014, 34(9): 3555-3574. doi: 10.3934/dcds.2014.34.3555

An averaging theorem for nonlinear Schrödinger equations with small nonlinearities

1. 

Centre Mathémathiques Laurent Schwartz, École Polytechnique, Palaiseau, 91125, France

Received  July 2013 Revised  December 2013 Published  March 2014

Consider nonlinear Schrödinger equations with small nonlinearities \[\frac{d}{dt}u+i(-\triangle u+V(x)u)=\epsilon \mathcal{P}(\triangle u,\nabla u,u,x),           x\in \mathbb{T}^d.                     (*)\] Let $\{\zeta_1(x),\zeta_2(x),\dots\}$ be the $L_2$-basis formed by eigenfunctions of the operator $-\triangle +V(x)$. For any complex function $u(x)$, write it as $ u(x) = \sum_{k≥1} v_k \zeta_k (x)$ and set $I_k(u)=\frac{1}{2}|v_k|^2$. Then for any solution $u(t,x)$ of the linear equation $(*)_{\epsilon=0}$ we have $I(u(t,\cdot))=const$. In this work it is proved that if $(*)$ is well posed on time-intervals $t≲\epsilon^{-1}$ and satisfies there some mild a-priori assumptions, then for any its solution $u^{\epsilon}(t,x)$, the limiting behavior of the curve $I(u^{\epsilon}(t,\cdot))$ on time intervals of order $\epsilon^{-1}$, as $\epsilon\to0$, can be uniquely characterized by solutions of a certain well-posed effective equation.
Citation: Guan Huang. An averaging theorem for nonlinear Schrödinger equations with small nonlinearities. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3555-3574. doi: 10.3934/dcds.2014.34.3555
References:
[1]

D. Bambusi, Nekhoroshev theorem for small amplitude solutions in nonlinear Schrödinger equation, Math. Z., 230 (1999), 345-387. doi: 10.1007/PL00004696.

[2]

D. Bambusi, Galerkin averaging method and Poincaré normal form for some quasilnear PDEs, Ann. Scuola Norm. Sup. Pisa C1. Sci., IV (2005), 669-702.

[3]

D. Bambusi and B. Grebert, Forme normale pour NLS en dimension quelconque, Comptes Rendus Mathématique, 337 (2003), 409-414. doi: 10.1016/S1631-073X(03)00368-6.

[4]

V. Bogachev, Differentiable Measures and the Malliavin Calculus, American Mathematical Society, 2010.

[5]

V. Bogachev and I. Malofeev, On the absolute continuity of the distributions of smooth functions on infinite-dimensional spaces with measures, preprint, (2013).

[6]

J. Bourgain, On diffusion in high-dimensional Hamiltonian systems and PDEs, Journal d'Analyse Mathématique, 80 (2000), 1-35. doi: 10.1007/BF02791532.

[7]

G. Huang, An averaging theorem for a perturbed KdV equation, Nonlinearity, 26 (2013), 1599-1621. doi: 10.1088/0951-7715/26/6/1599.

[8]

T. Kappeler and S. Kuksin, Strong non-resonance of Schrédinger operators and an averaging theorem, Physica D, 86 (1995), 349-362. doi: 10.1016/0167-2789(95)00115-K.

[9]

S. Kuksin, Damped-driven KdV and effective equations for long-time behavior of its solutions, GAFA, 20 (2010), 1431-1463. doi: 10.1007/s00039-010-0103-6.

[10]

S. Kuksin, Weakly nonlinear stochastic CGL equations, Annales de l'insitut Henri Poincaré-Probabilité et Statistiques, 49 (2013), 915-1231. doi: 10.1214/11-AIHP482.

[11]

S. Kuksin and A. Piatnitski, Khasminskii-Whitham averaging for randomly perturbed KdV equation, J. Math. Pures Appl., 89 (2008), 400-428. doi: 10.1016/j.matpur.2007.12.003.

[12]

P. Lochak and C. Meunier, Multiphase Averaging for Classical Systems, Springer-Verlag, 1988. doi: 10.1007/978-1-4612-1044-3.

[13]

J. Pöschel, On Nekhoroshev estimates for a nonlinear Schrödinger equation and a theorem by Bambusi, Nonlinearity, 12 (1999), 1587-1600. doi: 10.1088/0951-7715/12/6/310.

[14]

T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, de Gruyter, 1996. doi: 10.1515/9783110812411.

[15]

H. Whitney, Differentiable even functions, Duke Math. Journal, 10 (1943), 159-160. doi: 10.1215/S0012-7094-43-01015-4.

show all references

References:
[1]

D. Bambusi, Nekhoroshev theorem for small amplitude solutions in nonlinear Schrödinger equation, Math. Z., 230 (1999), 345-387. doi: 10.1007/PL00004696.

[2]

D. Bambusi, Galerkin averaging method and Poincaré normal form for some quasilnear PDEs, Ann. Scuola Norm. Sup. Pisa C1. Sci., IV (2005), 669-702.

[3]

D. Bambusi and B. Grebert, Forme normale pour NLS en dimension quelconque, Comptes Rendus Mathématique, 337 (2003), 409-414. doi: 10.1016/S1631-073X(03)00368-6.

[4]

V. Bogachev, Differentiable Measures and the Malliavin Calculus, American Mathematical Society, 2010.

[5]

V. Bogachev and I. Malofeev, On the absolute continuity of the distributions of smooth functions on infinite-dimensional spaces with measures, preprint, (2013).

[6]

J. Bourgain, On diffusion in high-dimensional Hamiltonian systems and PDEs, Journal d'Analyse Mathématique, 80 (2000), 1-35. doi: 10.1007/BF02791532.

[7]

G. Huang, An averaging theorem for a perturbed KdV equation, Nonlinearity, 26 (2013), 1599-1621. doi: 10.1088/0951-7715/26/6/1599.

[8]

T. Kappeler and S. Kuksin, Strong non-resonance of Schrédinger operators and an averaging theorem, Physica D, 86 (1995), 349-362. doi: 10.1016/0167-2789(95)00115-K.

[9]

S. Kuksin, Damped-driven KdV and effective equations for long-time behavior of its solutions, GAFA, 20 (2010), 1431-1463. doi: 10.1007/s00039-010-0103-6.

[10]

S. Kuksin, Weakly nonlinear stochastic CGL equations, Annales de l'insitut Henri Poincaré-Probabilité et Statistiques, 49 (2013), 915-1231. doi: 10.1214/11-AIHP482.

[11]

S. Kuksin and A. Piatnitski, Khasminskii-Whitham averaging for randomly perturbed KdV equation, J. Math. Pures Appl., 89 (2008), 400-428. doi: 10.1016/j.matpur.2007.12.003.

[12]

P. Lochak and C. Meunier, Multiphase Averaging for Classical Systems, Springer-Verlag, 1988. doi: 10.1007/978-1-4612-1044-3.

[13]

J. Pöschel, On Nekhoroshev estimates for a nonlinear Schrödinger equation and a theorem by Bambusi, Nonlinearity, 12 (1999), 1587-1600. doi: 10.1088/0951-7715/12/6/310.

[14]

T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, de Gruyter, 1996. doi: 10.1515/9783110812411.

[15]

H. Whitney, Differentiable even functions, Duke Math. Journal, 10 (1943), 159-160. doi: 10.1215/S0012-7094-43-01015-4.

[1]

Zhijian Yang, Ke Li. Longtime dynamics for an elastic waveguide model. Conference Publications, 2013, 2013 (special) : 797-806. doi: 10.3934/proc.2013.2013.797

[2]

Sébastien Guisset. Angular moments models for rarefied gas dynamics. Numerical comparisons with kinetic and Navier-Stokes equations. Kinetic and Related Models, 2020, 13 (4) : 739-758. doi: 10.3934/krm.2020025

[3]

Gongwei Liu, Baowei Feng, Xinguang Yang. Longtime dynamics for a type of suspension bridge equation with past history and time delay. Communications on Pure and Applied Analysis, 2020, 19 (10) : 4995-5013. doi: 10.3934/cpaa.2020224

[4]

Michele Colturato. Well-posedness and longtime behavior for a singular phase field system with perturbed phase dynamics. Evolution Equations and Control Theory, 2018, 7 (2) : 217-245. doi: 10.3934/eect.2018011

[5]

Mouhamadou Samsidy Goudiaby, Ababacar Diagne, Leon Matar Tine. Longtime behavior of a second order finite element scheme simulating the kinematic effects in liquid crystal dynamics. Communications on Pure and Applied Analysis, 2021, 20 (10) : 3499-3514. doi: 10.3934/cpaa.2021116

[6]

Jianjun Yuan. Derivation of the Quintic NLS from many-body quantum dynamics in $T^2$. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1941-1960. doi: 10.3934/cpaa.2015.14.1941

[7]

M. Grasselli, Vittorino Pata, Giovanni Prouse. Longtime behavior of a viscoelastic Timoshenko beam. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 337-348. doi: 10.3934/dcds.2004.10.337

[8]

Messoud A. Efendiev, Sergey Zelik, Hermann J. Eberl. Existence and longtime behavior of a biofilm model. Communications on Pure and Applied Analysis, 2009, 8 (2) : 509-531. doi: 10.3934/cpaa.2009.8.509

[9]

M. Grasselli, Vittorino Pata. Longtime behavior of a homogenized model in viscoelastodynamics. Discrete and Continuous Dynamical Systems, 1998, 4 (2) : 339-358. doi: 10.3934/dcds.1998.4.339

[10]

Rémi Carles, Erwan Faou. Energy cascades for NLS on the torus. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 2063-2077. doi: 10.3934/dcds.2012.32.2063

[11]

S. L. Ma'u, P. Ramankutty. An averaging method for the Helmholtz equation. Conference Publications, 2003, 2003 (Special) : 604-609. doi: 10.3934/proc.2003.2003.604

[12]

Natalia Skripnik. Averaging of fuzzy integral equations. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 1999-2010. doi: 10.3934/dcdsb.2017118

[13]

Ciprian G. Gal, Maurizio Grasselli. Longtime behavior of nonlocal Cahn-Hilliard equations. Discrete and Continuous Dynamical Systems, 2014, 34 (1) : 145-179. doi: 10.3934/dcds.2014.34.145

[14]

Zhijian Yang, Zhiming Liu, Na Feng. Longtime behavior of the semilinear wave equation with gentle dissipation. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6557-6580. doi: 10.3934/dcds.2016084

[15]

Patrick Cummings, C. Eugene Wayne. Modified energy functionals and the NLS approximation. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1295-1321. doi: 10.3934/dcds.2017054

[16]

Janusz Mierczyński. Averaging in random systems of nonnegative matrices. Conference Publications, 2015, 2015 (special) : 835-840. doi: 10.3934/proc.2015.0835

[17]

Jinxin Xue. Continuous averaging proof of the Nekhoroshev theorem. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3827-3855. doi: 10.3934/dcds.2015.35.3827

[18]

Andrej V. Plotnikov, Tatyana A. Komleva, Liliya I. Plotnikova. The averaging of fuzzy hyperbolic differential inclusions. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 1987-1998. doi: 10.3934/dcdsb.2017117

[19]

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

[20]

Ciprian G. Gal, Maurizio Grasselli. Longtime behavior for a model of homogeneous incompressible two-phase flows. Discrete and Continuous Dynamical Systems, 2010, 28 (1) : 1-39. doi: 10.3934/dcds.2010.28.1

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (73)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]