June  2019, 39(6): 3239-3264. doi: 10.3934/dcds.2019134

Uniform Strichartz estimates on the lattice

1. 

Department of Mathematics, Chung-Ang University, Seoul 06974, Republic of Korea

2. 

Korea Institute for Advanced Study, Seoul 20455, Korea

3. 

Institute of Pure and Applied Mathematics, Chonbuk National University, Jeonju 54896, Korea

* Corresponding author: Changhun Yang

Received  June 2018 Published  February 2019

Fund Project: This Research of the first author was supported by the Chung-Ang University Research Grants in 2018. The second author was supported in part by Samsung Science and Technology Foundation under Project Number SSTF-BA1702-02

In this paper, we investigate Strichartz estimates for discrete linear Schrödinger and discrete linear Klein-Gordon equations on a lattice $ h\mathbb{Z}^d $ with $ h>0 $, where $ h $ is the distance between two adjacent lattice points. As for fixed $ h>0 $, Strichartz estimates for discrete Schrödinger and one-dimensional discrete Klein-Gordon equations are established by Stefanov-Kevrekidis [21]. Our main result shows that such inequalities hold uniformly in $ h\in(0,1] $ with additional fractional derivatives on the right hand side. As an application, we obtain local well-posedness of a discrete nonlinear Schrödinger equation with a priori bounds independent of $ h $. The theorems and the harmonic analysis tools developed in this paper would be useful in the study of the continuum limit $ h\to 0 $ for discrete models, including our forthcoming work [7] where strong convergence for a discrete nonlinear Schrödinger equation is addressed.

Citation: Younghun Hong, Changhun Yang. Uniform Strichartz estimates on the lattice. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3239-3264. doi: 10.3934/dcds.2019134
References:
[1]

F. CataliottiS. BurgerC. FortP. MaddaloniF. MinardiA. TrombettoniA. Smerzi and M. Inguscio, Josephson junction arrays with bose-einstein condensates, Science, 293 (2001), 843-846. doi: 10.1126/science.1062612. Google Scholar

[2]

F. Cataliotti, L. Fallani, F. Ferlaino, C. Fort, P. Maddaloni and M. Inguscio, Superfluid current disruption in a chain of weakly coupled Bose–Einstein condensates, New Journal of Physics, 5 (2003), 71. doi: 10.1088/1367-2630/5/1/371. Google Scholar

[3]

S. Chatterjee, Invariant measures and the soliton resolution conjecture, Comm. Pure Appl. Math., 67 (2014), 1737-1842. doi: 10.1002/cpa.21501. Google Scholar

[4]

J. Duoandikoetxea, Fourier Analysis, vol. 29 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2001, Translated and revised from the 1995 Spanish original by David Cruz-Uribe. Google Scholar

[5]

H. EisenbergR. MorandottiY. SilberbergJ. ArnoldG. Pennelli and J. Aitchison, Optical discrete solitons in waveguide arrays. i. soliton formation, JOSA B, 19 (2002), 2938-2944. doi: 10.1364/JOSAB.19.002938. Google Scholar

[6]

U. Haagerup, The best constants in the khintchine inequality, Studia Mathematica, 70 (1981), 231-283. doi: 10.4064/sm-70-3-231-283. Google Scholar

[7]

Y. Hong and C. Yang, Strong Convergence for Discrete Nonlinear Schrödinger equations in the Continuum Limit, arXiv: 1806.07542.Google Scholar

[8]

L. I. Ignat, Fully discrete schemes for the Schrödinger equation. Dispersive properties, Math. Models Methods Appl. Sci., 17 (2007), 567-591. doi: 10.1142/S0218202507002029. Google Scholar

[9]

L. I. Ignat and E. Zuazua, Dispersive properties of a viscous numerical scheme for the Schrödinger equation, C. R. Math. Acad. Sci. Paris, 340 (2005), 529-534. doi: 10.1016/j.crma.2005.02.024. Google Scholar

[10]

L. I. Ignat and E. Zuazua, A two-grid approximation scheme for nonlinear Schrödinger equations: dispersive properties and convergence, C. R. Math. Acad. Sci. Paris, 341 (2005), 381-386. doi: 10.1016/j.crma.2005.07.018. Google Scholar

[11]

L. I. Ignat and E. Zuazua, Convergence of a two-grid algorithm for the control of the wave equation, J. Eur. Math. Soc. (JEMS), 11 (2009), 351-391. doi: 10.4171/JEMS/153. Google Scholar

[12]

L. I. Ignat and E. Zuazua, Numerical dispersive schemes for the nonlinear Schrödinger equation, SIAM J. Numer. Anal., 47 (2009), 1366-1390. doi: 10.1137/070683787. Google Scholar

[13]

L. I. Ignat and E. Zuazua, Convergence rates for dispersive approximation schemes to nonlinear Schrödinger equations, J. Math. Pures Appl. (9), 98 (2012), 479-517. doi: 10.1016/j.matpur.2012.01.001. Google Scholar

[14]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980, URL http://muse.jhu.edu/journals/american_journal_of_mathematics/v120/120.5keel.pdf. doi: 10.1353/ajm.1998.0039. Google Scholar

[15]

P. G. Kevrekidis, The Discrete Nonlinear Schrödinger Equation: Mathematical Analysis, Numerical Computations and Physical Perspectives, vol. 232, Springer Science & Business Media, 2009. doi: 10.1007/978-3-540-89199-4. Google Scholar

[16]

P. KevrekidisK. Rasmussen and A. Bishop, The discrete nonlinear Schrödinger equation: A survey of recent results, International Journal of Modern Physics B, 15 (2001), 2833-2900. doi: 10.1142/S0217979201007105. Google Scholar

[17]

K. KirkpatrickE. Lenzmann and G. Staffilani, On the continuum limit for discrete NLS with long-range lattice interactions, Comm. Math. Phys., 317 (2013), 563-591. doi: 10.1007/s00220-012-1621-x. Google Scholar

[18]

S. Mingaleev, P. Christiansen, Y. Gaididei, M. Johansson and K. Rasmussen, Models for energy and charge transport and storage in biomolecules, Journal of Biological Physics, 25 (1999), 41-63, URL https://doi.org/10.1023/A:1005152704984.Google Scholar

[19]

U. PeschelR. MorandottiJ. M. ArnoldJ. S. AitchisonH. S. EisenbergY. SilberbergT. Pertsch and F. Lederer, Optical discrete solitons in waveguide arrays. 2. dynamic properties, JOSA B, 19 (2002), 2637-2644. Google Scholar

[20]

M. Peyrard and A. R. Bishop, Statistical mechanics of a nonlinear model for dna denaturation, Physical Review Letters, 62 (1989), 2755. doi: 10.1103/PhysRevLett.62.2755. Google Scholar

[21]

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

[22]

A. A. SukhorukovY. S. KivsharH. S. Eisenberg and Y. Silberberg, Spatial optical solitons in waveguide arrays, IEEE Journal of Quantum Electronics, 39 (2003), 31-50. doi: 10.1109/JQE.2002.806184. Google Scholar

[23]

A. Ustinov, M. Cirillo and B. Malomed, Fluxon dynamics in one-dimensional josephson-junction arrays, Physical Review B, 47 (1993), 8357. doi: 10.1103/PhysRevB.47.8357. Google Scholar

show all references

References:
[1]

F. CataliottiS. BurgerC. FortP. MaddaloniF. MinardiA. TrombettoniA. Smerzi and M. Inguscio, Josephson junction arrays with bose-einstein condensates, Science, 293 (2001), 843-846. doi: 10.1126/science.1062612. Google Scholar

[2]

F. Cataliotti, L. Fallani, F. Ferlaino, C. Fort, P. Maddaloni and M. Inguscio, Superfluid current disruption in a chain of weakly coupled Bose–Einstein condensates, New Journal of Physics, 5 (2003), 71. doi: 10.1088/1367-2630/5/1/371. Google Scholar

[3]

S. Chatterjee, Invariant measures and the soliton resolution conjecture, Comm. Pure Appl. Math., 67 (2014), 1737-1842. doi: 10.1002/cpa.21501. Google Scholar

[4]

J. Duoandikoetxea, Fourier Analysis, vol. 29 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2001, Translated and revised from the 1995 Spanish original by David Cruz-Uribe. Google Scholar

[5]

H. EisenbergR. MorandottiY. SilberbergJ. ArnoldG. Pennelli and J. Aitchison, Optical discrete solitons in waveguide arrays. i. soliton formation, JOSA B, 19 (2002), 2938-2944. doi: 10.1364/JOSAB.19.002938. Google Scholar

[6]

U. Haagerup, The best constants in the khintchine inequality, Studia Mathematica, 70 (1981), 231-283. doi: 10.4064/sm-70-3-231-283. Google Scholar

[7]

Y. Hong and C. Yang, Strong Convergence for Discrete Nonlinear Schrödinger equations in the Continuum Limit, arXiv: 1806.07542.Google Scholar

[8]

L. I. Ignat, Fully discrete schemes for the Schrödinger equation. Dispersive properties, Math. Models Methods Appl. Sci., 17 (2007), 567-591. doi: 10.1142/S0218202507002029. Google Scholar

[9]

L. I. Ignat and E. Zuazua, Dispersive properties of a viscous numerical scheme for the Schrödinger equation, C. R. Math. Acad. Sci. Paris, 340 (2005), 529-534. doi: 10.1016/j.crma.2005.02.024. Google Scholar

[10]

L. I. Ignat and E. Zuazua, A two-grid approximation scheme for nonlinear Schrödinger equations: dispersive properties and convergence, C. R. Math. Acad. Sci. Paris, 341 (2005), 381-386. doi: 10.1016/j.crma.2005.07.018. Google Scholar

[11]

L. I. Ignat and E. Zuazua, Convergence of a two-grid algorithm for the control of the wave equation, J. Eur. Math. Soc. (JEMS), 11 (2009), 351-391. doi: 10.4171/JEMS/153. Google Scholar

[12]

L. I. Ignat and E. Zuazua, Numerical dispersive schemes for the nonlinear Schrödinger equation, SIAM J. Numer. Anal., 47 (2009), 1366-1390. doi: 10.1137/070683787. Google Scholar

[13]

L. I. Ignat and E. Zuazua, Convergence rates for dispersive approximation schemes to nonlinear Schrödinger equations, J. Math. Pures Appl. (9), 98 (2012), 479-517. doi: 10.1016/j.matpur.2012.01.001. Google Scholar

[14]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980, URL http://muse.jhu.edu/journals/american_journal_of_mathematics/v120/120.5keel.pdf. doi: 10.1353/ajm.1998.0039. Google Scholar

[15]

P. G. Kevrekidis, The Discrete Nonlinear Schrödinger Equation: Mathematical Analysis, Numerical Computations and Physical Perspectives, vol. 232, Springer Science & Business Media, 2009. doi: 10.1007/978-3-540-89199-4. Google Scholar

[16]

P. KevrekidisK. Rasmussen and A. Bishop, The discrete nonlinear Schrödinger equation: A survey of recent results, International Journal of Modern Physics B, 15 (2001), 2833-2900. doi: 10.1142/S0217979201007105. Google Scholar

[17]

K. KirkpatrickE. Lenzmann and G. Staffilani, On the continuum limit for discrete NLS with long-range lattice interactions, Comm. Math. Phys., 317 (2013), 563-591. doi: 10.1007/s00220-012-1621-x. Google Scholar

[18]

S. Mingaleev, P. Christiansen, Y. Gaididei, M. Johansson and K. Rasmussen, Models for energy and charge transport and storage in biomolecules, Journal of Biological Physics, 25 (1999), 41-63, URL https://doi.org/10.1023/A:1005152704984.Google Scholar

[19]

U. PeschelR. MorandottiJ. M. ArnoldJ. S. AitchisonH. S. EisenbergY. SilberbergT. Pertsch and F. Lederer, Optical discrete solitons in waveguide arrays. 2. dynamic properties, JOSA B, 19 (2002), 2637-2644. Google Scholar

[20]

M. Peyrard and A. R. Bishop, Statistical mechanics of a nonlinear model for dna denaturation, Physical Review Letters, 62 (1989), 2755. doi: 10.1103/PhysRevLett.62.2755. Google Scholar

[21]

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

[22]

A. A. SukhorukovY. S. KivsharH. S. Eisenberg and Y. Silberberg, Spatial optical solitons in waveguide arrays, IEEE Journal of Quantum Electronics, 39 (2003), 31-50. doi: 10.1109/JQE.2002.806184. Google Scholar

[23]

A. Ustinov, M. Cirillo and B. Malomed, Fluxon dynamics in one-dimensional josephson-junction arrays, Physical Review B, 47 (1993), 8357. doi: 10.1103/PhysRevB.47.8357. Google Scholar

Figure 1.  Strichartz estimates $(q,r)$ pair for $d = 3$
Figure 2.  Degenerate point in Fourier side for $ d = 2 $
Figure 3.  Domain in lattice and Fourier side for $ d = 2 $
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