# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - A, 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. doi: 10.1007/PL00004696. Google Scholar [2] D. Bambusi, Galerkin averaging method and Poincaré normal form for some quasilnear PDEs,, Ann. Scuola Norm. Sup. Pisa C1. Sci., IV (2005), 669. Google Scholar [3] D. Bambusi and B. Grebert, Forme normale pour NLS en dimension quelconque,, Comptes Rendus Mathématique, 337 (2003), 409. doi: 10.1016/S1631-073X(03)00368-6. Google Scholar [4] V. Bogachev, Differentiable Measures and the Malliavin Calculus,, American Mathematical Society, (2010). Google Scholar [5] V. Bogachev and I. Malofeev, On the absolute continuity of the distributions of smooth functions on infinite-dimensional spaces with measures,, preprint, (2013). Google Scholar [6] J. Bourgain, On diffusion in high-dimensional Hamiltonian systems and PDEs,, Journal d'Analyse Mathématique, 80 (2000), 1. doi: 10.1007/BF02791532. Google Scholar [7] G. Huang, An averaging theorem for a perturbed KdV equation,, Nonlinearity, 26 (2013), 1599. doi: 10.1088/0951-7715/26/6/1599. Google Scholar [8] T. Kappeler and S. Kuksin, Strong non-resonance of Schrédinger operators and an averaging theorem,, Physica D, 86 (1995), 349. doi: 10.1016/0167-2789(95)00115-K. Google Scholar [9] S. Kuksin, Damped-driven KdV and effective equations for long-time behavior of its solutions,, GAFA, 20 (2010), 1431. doi: 10.1007/s00039-010-0103-6. Google Scholar [10] S. Kuksin, Weakly nonlinear stochastic CGL equations,, Annales de l'insitut Henri Poincaré-Probabilité et Statistiques, 49 (2013), 915. doi: 10.1214/11-AIHP482. Google Scholar [11] S. Kuksin and A. Piatnitski, Khasminskii-Whitham averaging for randomly perturbed KdV equation,, J. Math. Pures Appl., 89 (2008), 400. doi: 10.1016/j.matpur.2007.12.003. Google Scholar [12] P. Lochak and C. Meunier, Multiphase Averaging for Classical Systems,, Springer-Verlag, (1988). doi: 10.1007/978-1-4612-1044-3. Google Scholar [13] J. Pöschel, On Nekhoroshev estimates for a nonlinear Schrödinger equation and a theorem by Bambusi,, Nonlinearity, 12 (1999), 1587. doi: 10.1088/0951-7715/12/6/310. Google Scholar [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. Google Scholar [15] H. Whitney, Differentiable even functions,, Duke Math. Journal, 10 (1943), 159. doi: 10.1215/S0012-7094-43-01015-4. Google Scholar

show all references

##### References:
 [1] D. Bambusi, Nekhoroshev theorem for small amplitude solutions in nonlinear Schrödinger equation,, Math. Z., 230 (1999), 345. doi: 10.1007/PL00004696. Google Scholar [2] D. Bambusi, Galerkin averaging method and Poincaré normal form for some quasilnear PDEs,, Ann. Scuola Norm. Sup. Pisa C1. Sci., IV (2005), 669. Google Scholar [3] D. Bambusi and B. Grebert, Forme normale pour NLS en dimension quelconque,, Comptes Rendus Mathématique, 337 (2003), 409. doi: 10.1016/S1631-073X(03)00368-6. Google Scholar [4] V. Bogachev, Differentiable Measures and the Malliavin Calculus,, American Mathematical Society, (2010). Google Scholar [5] V. Bogachev and I. Malofeev, On the absolute continuity of the distributions of smooth functions on infinite-dimensional spaces with measures,, preprint, (2013). Google Scholar [6] J. Bourgain, On diffusion in high-dimensional Hamiltonian systems and PDEs,, Journal d'Analyse Mathématique, 80 (2000), 1. doi: 10.1007/BF02791532. Google Scholar [7] G. Huang, An averaging theorem for a perturbed KdV equation,, Nonlinearity, 26 (2013), 1599. doi: 10.1088/0951-7715/26/6/1599. Google Scholar [8] T. Kappeler and S. Kuksin, Strong non-resonance of Schrédinger operators and an averaging theorem,, Physica D, 86 (1995), 349. doi: 10.1016/0167-2789(95)00115-K. Google Scholar [9] S. Kuksin, Damped-driven KdV and effective equations for long-time behavior of its solutions,, GAFA, 20 (2010), 1431. doi: 10.1007/s00039-010-0103-6. Google Scholar [10] S. Kuksin, Weakly nonlinear stochastic CGL equations,, Annales de l'insitut Henri Poincaré-Probabilité et Statistiques, 49 (2013), 915. doi: 10.1214/11-AIHP482. Google Scholar [11] S. Kuksin and A. Piatnitski, Khasminskii-Whitham averaging for randomly perturbed KdV equation,, J. Math. Pures Appl., 89 (2008), 400. doi: 10.1016/j.matpur.2007.12.003. Google Scholar [12] P. Lochak and C. Meunier, Multiphase Averaging for Classical Systems,, Springer-Verlag, (1988). doi: 10.1007/978-1-4612-1044-3. Google Scholar [13] J. Pöschel, On Nekhoroshev estimates for a nonlinear Schrödinger equation and a theorem by Bambusi,, Nonlinearity, 12 (1999), 1587. doi: 10.1088/0951-7715/12/6/310. Google Scholar [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. Google Scholar [15] H. Whitney, Differentiable even functions,, Duke Math. Journal, 10 (1943), 159. doi: 10.1215/S0012-7094-43-01015-4. Google Scholar
 [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] Michele Colturato. Well-posedness and longtime behavior for a singular phase field system with perturbed phase dynamics. Evolution Equations & Control Theory, 2018, 7 (2) : 217-245. doi: 10.3934/eect.2018011 [3] Jianjun Yuan. Derivation of the Quintic NLS from many-body quantum dynamics in $T^2$. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1941-1960. doi: 10.3934/cpaa.2015.14.1941 [4] M. Grasselli, Vittorino Pata, Giovanni Prouse. Longtime behavior of a viscoelastic Timoshenko beam. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 337-348. doi: 10.3934/dcds.2004.10.337 [5] Messoud A. Efendiev, Sergey Zelik, Hermann J. Eberl. Existence and longtime behavior of a biofilm model. Communications on Pure & Applied Analysis, 2009, 8 (2) : 509-531. doi: 10.3934/cpaa.2009.8.509 [6] M. Grasselli, Vittorino Pata. Longtime behavior of a homogenized model in viscoelastodynamics. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 339-358. doi: 10.3934/dcds.1998.4.339 [7] Rémi Carles, Erwan Faou. Energy cascades for NLS on the torus. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2063-2077. doi: 10.3934/dcds.2012.32.2063 [8] Ciprian G. Gal, Maurizio Grasselli. Longtime behavior of nonlocal Cahn-Hilliard equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 145-179. doi: 10.3934/dcds.2014.34.145 [9] Zhijian Yang, Zhiming Liu, Na Feng. Longtime behavior of the semilinear wave equation with gentle dissipation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6557-6580. doi: 10.3934/dcds.2016084 [10] 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 [11] Natalia Skripnik. Averaging of fuzzy integral equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1999-2010. doi: 10.3934/dcdsb.2017118 [12] Patrick Cummings, C. Eugene Wayne. Modified energy functionals and the NLS approximation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1295-1321. doi: 10.3934/dcds.2017054 [13] Ciprian G. Gal, Maurizio Grasselli. Longtime behavior for a model of homogeneous incompressible two-phase flows. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 1-39. doi: 10.3934/dcds.2010.28.1 [14] Doan Duy Hai, Atsushi Yagi. Longtime behavior of solutions to chemotaxis-proliferation model with three variables. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 3957-3974. doi: 10.3934/dcds.2012.32.3957 [15] Janusz Mierczyński. Averaging in random systems of nonnegative matrices. Conference Publications, 2015, 2015 (special) : 835-840. doi: 10.3934/proc.2015.0835 [16] Jinxin Xue. Continuous averaging proof of the Nekhoroshev theorem. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3827-3855. doi: 10.3934/dcds.2015.35.3827 [17] Andrej V. Plotnikov, Tatyana A. Komleva, Liliya I. Plotnikova. The averaging of fuzzy hyperbolic differential inclusions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1987-1998. doi: 10.3934/dcdsb.2017117 [18] 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 [19] Vassilis Rothos. Subharmonic bifurcations of localized solutions of a discrete NLS equation. Conference Publications, 2005, 2005 (Special) : 756-767. doi: 10.3934/proc.2005.2005.756 [20] A. Adam Azzam. Scattering for the two dimensional NLS with (full) exponential nonlinearity. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1071-1101. doi: 10.3934/cpaa.2018052

2018 Impact Factor: 1.143