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On substitution tilings and Delone sets without finite local complexity
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 |
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.
References:
[1] |
M. J. Ablowitz, B. Prinari and A. D. Trubatch, Discrete and Continuous Nonlinear Schrödinger Systems, Cambridge University Press, Cambridge, 2004.
![]() ![]() |
[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. |
[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. |
[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. |
[5] |
J. Bernier and E. Faou, Existence and stability of traveling waves for discrete nonlinear Schrödinger equations over long times, preprint, arXiv: 1805.03578. |
[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. |
[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. |
[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. |
[9] |
D. Furihata and T. Matsuo,
Discrete variational derivative method—a structure preserving numerical method for partial differential equations, Sūgaku, 66 (2014), 135-156.
|
[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. |
[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. |
[12] |
P. G. Kevrekidis, The Discrete Nonlinear Schrödinger Equation, Springer-Verlag, Berlin, 2009.
doi: 10.1007/978-3-540-89199-4. |
[13] |
J. Peyrière, Convolution, Séries et Intégrales de Fourier, (French) Références Sciences. Ellipses, Paris, 2012. |
[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. |
[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. |
[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. |
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.
![]() ![]() |
[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. |
[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. |
[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. |
[5] |
J. Bernier and E. Faou, Existence and stability of traveling waves for discrete nonlinear Schrödinger equations over long times, preprint, arXiv: 1805.03578. |
[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. |
[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. |
[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. |
[9] |
D. Furihata and T. Matsuo,
Discrete variational derivative method—a structure preserving numerical method for partial differential equations, Sūgaku, 66 (2014), 135-156.
|
[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. |
[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. |
[12] |
P. G. Kevrekidis, The Discrete Nonlinear Schrödinger Equation, Springer-Verlag, Berlin, 2009.
doi: 10.1007/978-3-540-89199-4. |
[13] |
J. Peyrière, Convolution, Séries et Intégrales de Fourier, (French) Références Sciences. Ellipses, Paris, 2012. |
[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. |
[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. |
[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. |
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