# American Institute of Mathematical Sciences

June  2019, 39(6): 3179-3195. doi: 10.3934/dcds.2019131

## Bounds on the growth of high discrete Sobolev norms for the cubic discrete nonlinear Schrödinger equations on $h\mathbb{Z}$

 Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France

* Corresponding author: Joackim Bernier

Received  May 2018 Revised  November 2018 Published  February 2019

We consider the discrete nonlinear Schrödinger equations on a one dimensional lattice of mesh $h$, with a cubic focusing or defocusing nonlinearity. We prove a polynomial bound on the growth of the discrete Sobolev norms, uniformly with respect to the stepsize of the grid. This bound is based on a construction of higher modified energies.

Citation: Joackim Bernier. Bounds on the growth of high discrete Sobolev norms for the cubic discrete nonlinear Schrödinger equations on $h\mathbb{Z}$. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3179-3195. doi: 10.3934/dcds.2019131
##### References:
 [1] M. J. Ablowitz, B. Prinari and A. D. Trubatch, Discrete and Continuous Nonlinear Schrödinger Systems, Cambridge University Press, Cambridge, 2004. Google Scholar [2] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, U.S. Government Printing Office, Washington, D.C., 1964. Google Scholar [3] D. Bambusi, E. Faou and B. Grébert, Existence and stability of ground states for fully discrete approximations of the nonlinear Schrödinger equation, Numer. Math., 123 (2013), 461-492. doi: 10.1007/s00211-012-0491-7. Google Scholar [4] D. Bambusi and T. Penati, Continuous approximation of breathers in one- and two-dimensional DNLS lattices, Nonlinearity, 23 (2010), 143-157. doi: 10.1088/0951-7715/23/1/008. Google Scholar [5] J. Bernier and E. Faou, Existence and stability of traveling waves for discrete nonlinear Schrödinger equations over long times, preprint, arXiv: 1805.03578.Google Scholar [6] J. Bourgain, On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE, Internat. Math. Res. Notices, (1996), 277–304. doi: 10.1155/S1073792896000207. Google Scholar [7] J. Bourgain, On growth of Sobolev norms in linear Schrödinger equations with smooth time dependent potential, J. Anal. Math., 77 (1999), 315-348. doi: 10.1007/BF02791265. Google Scholar [8] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Polynomial upper bounds for the orbital instability of the 1D cubic NLS below the energy norm, Discrete Contin. Dyn. Syst., 9 (2003), 31-54. doi: 10.3934/dcds.2003.9.31. Google Scholar [9] D. Furihata and T. Matsuo, Discrete variational derivative method—a structure preserving numerical method for partial differential equations, Sūgaku, 66 (2014), 135-156. Google Scholar [10] L. I. Ignat and E. Zuazua, Dispersive properties of numerical schemes for nonlinear Schrödinger equations, In Foundations of Computational Mathematics, Santander 2005, 331 (2006), 181–207. doi: 10.1017/CBO9780511721571.006. Google Scholar [11] M. Jenkinson and M. I. Weinstein., Onsite and offsite bound states of the discrete nonlinear Schrödinger equation and the Peierls-Nabarro barrier, Nonlinearity, 29 (2016), 27-86. doi: 10.1088/0951-7715/29/1/27. Google Scholar [12] P. G. Kevrekidis, The Discrete Nonlinear Schrödinger Equation, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-89199-4. Google Scholar [13] J. Peyrière, Convolution, Séries et Intégrales de Fourier, (French) Références Sciences. Ellipses, Paris, 2012. Google Scholar [14] V. Sohinger, Bounds on the growth of high Sobolev norms of solutions to nonlinear Schrödinger equations on ${\mathbb{R}}$, Indiana Univ. Math. J., 60 (2011), 1487-1516. doi: 10.1512/iumj.2011.60.4399. Google Scholar [15] G. Staffilani, On the growth of high Sobolev norms of solutions for KdV and Schrödinger equations, Duke Math. J., 86 (1997), 109-142. doi: 10.1215/S0012-7094-97-08604-X. Google Scholar [16] A. Stefanov and P. G. Kevrekidis, Asymptotic behaviour of small solutions for the discrete nonlinear Schrödinger and Klein-Gordon equations, Nonlinearity, 18 (2005), 1841-1857. doi: 10.1088/0951-7715/18/4/022. Google Scholar

show all references

##### References:
 [1] M. J. Ablowitz, B. Prinari and A. D. Trubatch, Discrete and Continuous Nonlinear Schrödinger Systems, Cambridge University Press, Cambridge, 2004. Google Scholar [2] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, U.S. Government Printing Office, Washington, D.C., 1964. Google Scholar [3] D. Bambusi, E. Faou and B. Grébert, Existence and stability of ground states for fully discrete approximations of the nonlinear Schrödinger equation, Numer. Math., 123 (2013), 461-492. doi: 10.1007/s00211-012-0491-7. Google Scholar [4] D. Bambusi and T. Penati, Continuous approximation of breathers in one- and two-dimensional DNLS lattices, Nonlinearity, 23 (2010), 143-157. doi: 10.1088/0951-7715/23/1/008. Google Scholar [5] J. Bernier and E. Faou, Existence and stability of traveling waves for discrete nonlinear Schrödinger equations over long times, preprint, arXiv: 1805.03578.Google Scholar [6] J. Bourgain, On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE, Internat. Math. Res. Notices, (1996), 277–304. doi: 10.1155/S1073792896000207. Google Scholar [7] J. Bourgain, On growth of Sobolev norms in linear Schrödinger equations with smooth time dependent potential, J. Anal. Math., 77 (1999), 315-348. doi: 10.1007/BF02791265. Google Scholar [8] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Polynomial upper bounds for the orbital instability of the 1D cubic NLS below the energy norm, Discrete Contin. Dyn. Syst., 9 (2003), 31-54. doi: 10.3934/dcds.2003.9.31. Google Scholar [9] D. Furihata and T. Matsuo, Discrete variational derivative method—a structure preserving numerical method for partial differential equations, Sūgaku, 66 (2014), 135-156. Google Scholar [10] L. I. Ignat and E. Zuazua, Dispersive properties of numerical schemes for nonlinear Schrödinger equations, In Foundations of Computational Mathematics, Santander 2005, 331 (2006), 181–207. doi: 10.1017/CBO9780511721571.006. Google Scholar [11] M. Jenkinson and M. I. Weinstein., Onsite and offsite bound states of the discrete nonlinear Schrödinger equation and the Peierls-Nabarro barrier, Nonlinearity, 29 (2016), 27-86. doi: 10.1088/0951-7715/29/1/27. Google Scholar [12] P. G. Kevrekidis, The Discrete Nonlinear Schrödinger Equation, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-89199-4. Google Scholar [13] J. Peyrière, Convolution, Séries et Intégrales de Fourier, (French) Références Sciences. Ellipses, Paris, 2012. Google Scholar [14] V. Sohinger, Bounds on the growth of high Sobolev norms of solutions to nonlinear Schrödinger equations on ${\mathbb{R}}$, Indiana Univ. Math. J., 60 (2011), 1487-1516. doi: 10.1512/iumj.2011.60.4399. Google Scholar [15] G. Staffilani, On the growth of high Sobolev norms of solutions for KdV and Schrödinger equations, Duke Math. J., 86 (1997), 109-142. doi: 10.1215/S0012-7094-97-08604-X. Google Scholar [16] A. Stefanov and P. G. Kevrekidis, Asymptotic behaviour of small solutions for the discrete nonlinear Schrödinger and Klein-Gordon equations, Nonlinearity, 18 (2005), 1841-1857. doi: 10.1088/0951-7715/18/4/022. Google Scholar
 [1] Myeongju Chae, Soonsik Kwon. The stability of nonlinear Schrödinger equations with a potential in high Sobolev norms revisited. Communications on Pure & Applied Analysis, 2016, 15 (2) : 341-365. doi: 10.3934/cpaa.2016.15.341 [2] F. Catoire, W. M. Wang. Bounds on Sobolev norms for the defocusing nonlinear Schrödinger equation on general flat tori. Communications on Pure & Applied Analysis, 2010, 9 (2) : 483-491. doi: 10.3934/cpaa.2010.9.483 [3] Seckin Demirbas. Local well-posedness for 2-D Schrödinger equation on irrational tori and bounds on Sobolev norms. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1517-1530. doi: 10.3934/cpaa.2017072 [4] François Genoud. Existence and stability of high frequency standing waves for a nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1229-1247. doi: 10.3934/dcds.2009.25.1229 [5] Vedran Sohinger. Bounds on the growth of high Sobolev norms of solutions to 2D Hartree equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3733-3771. doi: 10.3934/dcds.2012.32.3733 [6] Tetsu Mizumachi, Dmitry Pelinovsky. On the asymptotic stability of localized modes in the discrete nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 971-987. doi: 10.3934/dcdss.2012.5.971 [7] Nakao Hayashi, Pavel Naumkin. On the reduction of the modified Benjamin-Ono equation to the cubic derivative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 237-255. doi: 10.3934/dcds.2002.8.237 [8] Brenton LeMesurier. Modeling thermal effects on nonlinear wave motion in biopolymers by a stochastic discrete nonlinear Schrödinger equation with phase damping. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 317-327. doi: 10.3934/dcdss.2008.1.317 [9] D.G. deFigueiredo, Yanheng Ding. Solutions of a nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 563-584. doi: 10.3934/dcds.2002.8.563 [10] Miaomiao Niu, Zhongwei Tang. Least energy solutions for nonlinear Schrödinger equation involving the fractional Laplacian and critical growth. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3963-3987. doi: 10.3934/dcds.2017168 [11] J. Cuevas, J. C. Eilbeck, N. I. Karachalios. Thresholds for breather solutions of the discrete nonlinear Schrödinger equation with saturable and power nonlinearity. Discrete & Continuous Dynamical Systems - A, 2008, 21 (2) : 445-475. doi: 10.3934/dcds.2008.21.445 [12] In-Jee Jeong, Benoit Pausader. Discrete Schrödinger equation and ill-posedness for the Euler equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 281-293. doi: 10.3934/dcds.2017012 [13] Pavel I. Naumkin, Isahi Sánchez-Suárez. On the critical nongauge invariant nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 807-834. doi: 10.3934/dcds.2011.30.807 [14] Younghun Hong. Scattering for a nonlinear Schrödinger equation with a potential. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1571-1601. doi: 10.3934/cpaa.2016003 [15] Alexander Komech, Elena Kopylova, David Stuart. On asymptotic stability of solitons in a nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1063-1079. doi: 10.3934/cpaa.2012.11.1063 [16] Dario Bambusi, A. Carati, A. Ponno. The nonlinear Schrödinger equation as a resonant normal form. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 109-128. doi: 10.3934/dcdsb.2002.2.109 [17] Michal Fečkan, Vassilis M. Rothos. Travelling waves of forced discrete nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1129-1145. doi: 10.3934/dcdss.2011.4.1129 [18] Milena Stanislavova, Atanas Stefanov. Effective estimates of the higher Sobolev norms for the Kuramoto-Sivashinsky equation. Conference Publications, 2009, 2009 (Special) : 729-738. doi: 10.3934/proc.2009.2009.729 [19] Mohamad Darwich. On the $L^2$-critical nonlinear Schrödinger Equation with a nonlinear damping. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2377-2394. doi: 10.3934/cpaa.2014.13.2377 [20] Younghun Hong, Yannick Sire. On Fractional Schrödinger Equations in sobolev spaces. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2265-2282. doi: 10.3934/cpaa.2015.14.2265

2018 Impact Factor: 1.143