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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} $

Joackim Bernier ,

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.

Keywords: Discrete nonlinear Schrödinger equation, growth of high Sobolev norms, modified energies, Shannon interpolation, resonances.
Mathematics Subject Classification: Primary: 35Q55, 37K60; Secondary: 65P99.
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, 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

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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

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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

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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

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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

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