March  2014, 34(3): 991-1008. doi: 10.3934/dcds.2014.34.991

Numerical simulation of nonlinear dispersive quantization

1. 

School of Mathematics, University of Minnesota, Minneapolis, MN 55455, United States, United States

Received  November 2012 Revised  April 2013 Published  August 2013

When posed on a periodic domain in one space variable, linear dispersive evolution equations with integral polynomial dispersion relations exhibit strikingly different behaviors depending upon whether the time is rational or irrational relative to the length of the interval, thus producing the Talbot effect of dispersive quantization and fractalization. The goal here is to show that these remarkable phenomena extend to nonlinear dispersive evolution equations. We will present numerical simulations, based on operator splitting methods, of the nonlinear Schrödinger and Korteweg--deVries equations with step function initial data and periodic boundary conditions. For the integrable nonlinear Schrödinger equation, our observations have been rigorously confirmed in a recent paper of Erdoǧan and Tzirakis, [10].
Citation: Gong Chen, Peter J. Olver. Numerical simulation of nonlinear dispersive quantization. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 991-1008. doi: 10.3934/dcds.2014.34.991
References:
[1]

M. V. Berry, Quantum fractals in boxes, J. Phys. A, 29 (1996), 6617-6629. doi: 10.1088/0305-4470/29/20/016.

[2]

M. V. Berry, I. Marzoli and W. Schleich, Quantum carpets, carpets of light, Physics World, 14 (2001), 39-44.

[3]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal., 3 (1993), 107-156. doi: 10.1007/BF01896020.

[4]

J. Bourgain, Exponential sums and nonlinear Schrödinger equations, Geom. Funct. Anal., 3 (1993), 157-178. doi: 10.1007/BF01896021.

[5]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation, Geom. Funct. Anal., 3 (1993), 209-262. doi: 10.1007/BF01895688.

[6]

G. Chen and P. J. Olver, Dispersion of discontinuous periodic waves, Proc. Roy. Soc. London, 469 (2012), 20120407, 21 pp. doi: 10.1098/rspa.2012.0407.

[7]

C. M. Dafermos, "Hyperbolic Conservation Laws in Continuum Physics," 3rd ed., Springer Verlag, New York, 2010. doi: 10.1007/978-3-642-04048-1.

[8]

P. G. Drazin and R. S. Johnson, "Solitons: An Introduction," Cambridge University Press, Cambridge, 1989.

[9]

M. B. Erdoǧan and N. Tzirakis, Global smoothing for the periodic KdV evolution,, Internat. Math. Res. Notices, (). 

[10]

M. B. Erdoǧan and N. Tzirakis, Talbot effect for the cubic nonlinear Schrödinger equation on the torus, preprint, arXiv:1303.3604, 2013.

[11]

M. B. Erdoǧan, N. Tzirakis and V. Zharnitsky, Nearly linear dynamics of nonlinear dispersive waves, Physica D, 240 (2011), 1325-1333. doi: 10.1016/j.physd.2011.05.009.

[12]

H. Hanche-Olsen and H. Holden, The Kolmogorov-Riesz compactness theorem, Expo. Math., 28 (2010), 385-394. doi: 10.1016/j.exmath.2010.03.001.

[13]

H. Holden, K. H. Karlsen, K.-A. Lie and N. H. Risebro, "Splitting Methods for Partial Differential Equations with Rough Solutions: Analysis and MATLAB Programs," European Mathematical Society Publ., Zürich, 2010. doi: 10.4171/078.

[14]

H. Holden, K. H. Karlsen and N. H. Risebro, Operator splitting methods for generalized Korteweg-de Vries equations, J. Comput. Phys., 153 (1999), 203-222. doi: 10.1006/jcph.1999.6273.

[15]

H. Holden, K. H. Karlsen, N. H. Risebro and T. Tao, Operator splitting for the KdV equation, Math. Comp., 80 (2011), 821-846. doi: 10.1090/S0025-5718-2010-02402-0.

[16]

H. Holden, U. Koley and N. H. Risebro, Convergence of a fully discrete finite difference scheme for the Korteweg-de Vries equation, preprint, arXiv:1208.6410v1, 2012.

[17]

H. Holden, C. Lubich and N. H. Risebro, Operator splitting for partial differential equations with Burgers nonlinearity, Math. Comp., 82 (2013), 173-185. doi: 10.1090/S0025-5718-2012-02624-X.

[18]

L. Kapitanski and I. Rodnianski, Does a quantum particle know the time?, in"Emerging Applications of Number Theory" (eds. D. Hejhal, J. Friedman, M. C. Gutzwiller and A. M. Odlyzko), IMA Volumes in Mathematics and its Applications, 109, Springer Verlag, New York, (1999), 355-371. doi: 10.1007/978-1-4612-1544-8_14.

[19]

P. D. Lax and C. D. Levermore, The small dispersion limit of the Korteweg-deVries equation I, II, III, Commun. Pure Appl. Math., 36 (1983), 253-290. doi: 10.1002/cpa.3160360302.

[20]

C. Lubich, On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations, Math. Comp., 77 (2008), 2141-2153. doi: 10.1090/S0025-5718-08-02101-7.

[21]

K. D. T.-R. McLaughlin and N. J. E. Pitt, On ringing effects near jump discontinuities for periodic solutions to dispersive partial differential equations, preprint, arXiv:1107.1571v1, 2011.

[22]

P. J. Olver, Dispersive quantization, Amer. Math. Monthly, 117 (2010), 599-610. doi: 10.4169/000298910X496723.

[23]

K. I. Oskolkov, A class of I.M. Vinogradov's series and its applications in harmonic analysis, in "Progress in Approximation Theory," Springer Ser. Comput. Math., 19, Springer, New York, (1992), 353-402. doi: 10.1007/978-1-4612-2966-7_16.

[24]

K. Oskolkov, Schrödinger equation and oscillatory Hilbert transforms of second degree, J. Fourier Anal. Appl., 4 (1998), 341-356. doi: 10.1007/BF02476032.

[25]

I. Rodnianski, Fractal solutions of the Schrödinger equation, Contemp. Math., 255 (2000), 181-187. doi: 10.1090/conm/255/03981.

[26]

J. Smoller, "Shock Waves and Reaction-Diffusion Equations," 2nd edition, Springer-Verlag, New York, 1994.

[27]

H. F. Talbot, Facts related to optical science. No. IV, Philos. Mag., 9 (1836), 401-407. doi: 10.1080/14786443608649032.

[28]

M. Taylor, The Schrödinger equation on spheres, Pacific J. Math., 209 (2003), 145-155. doi: 10.2140/pjm.2003.209.145.

[29]

I. M. Vinogradov, "The Method of Trigonometrical Sums in the Theory of Numbers," Dover Publ., Mineola, NY, 2004.

[30]

G. B. Whitham, "Linear and Nonlinear Waves," John Wiley & Sons, New York, 1974.

[31]

Y. Zhou, Uniqueness of weak solution of the KdV equation, Internat. Math. Res. Notices, 1997 (1997), 271-283. doi: 10.1155/S1073792897000202.

show all references

References:
[1]

M. V. Berry, Quantum fractals in boxes, J. Phys. A, 29 (1996), 6617-6629. doi: 10.1088/0305-4470/29/20/016.

[2]

M. V. Berry, I. Marzoli and W. Schleich, Quantum carpets, carpets of light, Physics World, 14 (2001), 39-44.

[3]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal., 3 (1993), 107-156. doi: 10.1007/BF01896020.

[4]

J. Bourgain, Exponential sums and nonlinear Schrödinger equations, Geom. Funct. Anal., 3 (1993), 157-178. doi: 10.1007/BF01896021.

[5]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation, Geom. Funct. Anal., 3 (1993), 209-262. doi: 10.1007/BF01895688.

[6]

G. Chen and P. J. Olver, Dispersion of discontinuous periodic waves, Proc. Roy. Soc. London, 469 (2012), 20120407, 21 pp. doi: 10.1098/rspa.2012.0407.

[7]

C. M. Dafermos, "Hyperbolic Conservation Laws in Continuum Physics," 3rd ed., Springer Verlag, New York, 2010. doi: 10.1007/978-3-642-04048-1.

[8]

P. G. Drazin and R. S. Johnson, "Solitons: An Introduction," Cambridge University Press, Cambridge, 1989.

[9]

M. B. Erdoǧan and N. Tzirakis, Global smoothing for the periodic KdV evolution,, Internat. Math. Res. Notices, (). 

[10]

M. B. Erdoǧan and N. Tzirakis, Talbot effect for the cubic nonlinear Schrödinger equation on the torus, preprint, arXiv:1303.3604, 2013.

[11]

M. B. Erdoǧan, N. Tzirakis and V. Zharnitsky, Nearly linear dynamics of nonlinear dispersive waves, Physica D, 240 (2011), 1325-1333. doi: 10.1016/j.physd.2011.05.009.

[12]

H. Hanche-Olsen and H. Holden, The Kolmogorov-Riesz compactness theorem, Expo. Math., 28 (2010), 385-394. doi: 10.1016/j.exmath.2010.03.001.

[13]

H. Holden, K. H. Karlsen, K.-A. Lie and N. H. Risebro, "Splitting Methods for Partial Differential Equations with Rough Solutions: Analysis and MATLAB Programs," European Mathematical Society Publ., Zürich, 2010. doi: 10.4171/078.

[14]

H. Holden, K. H. Karlsen and N. H. Risebro, Operator splitting methods for generalized Korteweg-de Vries equations, J. Comput. Phys., 153 (1999), 203-222. doi: 10.1006/jcph.1999.6273.

[15]

H. Holden, K. H. Karlsen, N. H. Risebro and T. Tao, Operator splitting for the KdV equation, Math. Comp., 80 (2011), 821-846. doi: 10.1090/S0025-5718-2010-02402-0.

[16]

H. Holden, U. Koley and N. H. Risebro, Convergence of a fully discrete finite difference scheme for the Korteweg-de Vries equation, preprint, arXiv:1208.6410v1, 2012.

[17]

H. Holden, C. Lubich and N. H. Risebro, Operator splitting for partial differential equations with Burgers nonlinearity, Math. Comp., 82 (2013), 173-185. doi: 10.1090/S0025-5718-2012-02624-X.

[18]

L. Kapitanski and I. Rodnianski, Does a quantum particle know the time?, in"Emerging Applications of Number Theory" (eds. D. Hejhal, J. Friedman, M. C. Gutzwiller and A. M. Odlyzko), IMA Volumes in Mathematics and its Applications, 109, Springer Verlag, New York, (1999), 355-371. doi: 10.1007/978-1-4612-1544-8_14.

[19]

P. D. Lax and C. D. Levermore, The small dispersion limit of the Korteweg-deVries equation I, II, III, Commun. Pure Appl. Math., 36 (1983), 253-290. doi: 10.1002/cpa.3160360302.

[20]

C. Lubich, On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations, Math. Comp., 77 (2008), 2141-2153. doi: 10.1090/S0025-5718-08-02101-7.

[21]

K. D. T.-R. McLaughlin and N. J. E. Pitt, On ringing effects near jump discontinuities for periodic solutions to dispersive partial differential equations, preprint, arXiv:1107.1571v1, 2011.

[22]

P. J. Olver, Dispersive quantization, Amer. Math. Monthly, 117 (2010), 599-610. doi: 10.4169/000298910X496723.

[23]

K. I. Oskolkov, A class of I.M. Vinogradov's series and its applications in harmonic analysis, in "Progress in Approximation Theory," Springer Ser. Comput. Math., 19, Springer, New York, (1992), 353-402. doi: 10.1007/978-1-4612-2966-7_16.

[24]

K. Oskolkov, Schrödinger equation and oscillatory Hilbert transforms of second degree, J. Fourier Anal. Appl., 4 (1998), 341-356. doi: 10.1007/BF02476032.

[25]

I. Rodnianski, Fractal solutions of the Schrödinger equation, Contemp. Math., 255 (2000), 181-187. doi: 10.1090/conm/255/03981.

[26]

J. Smoller, "Shock Waves and Reaction-Diffusion Equations," 2nd edition, Springer-Verlag, New York, 1994.

[27]

H. F. Talbot, Facts related to optical science. No. IV, Philos. Mag., 9 (1836), 401-407. doi: 10.1080/14786443608649032.

[28]

M. Taylor, The Schrödinger equation on spheres, Pacific J. Math., 209 (2003), 145-155. doi: 10.2140/pjm.2003.209.145.

[29]

I. M. Vinogradov, "The Method of Trigonometrical Sums in the Theory of Numbers," Dover Publ., Mineola, NY, 2004.

[30]

G. B. Whitham, "Linear and Nonlinear Waves," John Wiley & Sons, New York, 1974.

[31]

Y. Zhou, Uniqueness of weak solution of the KdV equation, Internat. Math. Res. Notices, 1997 (1997), 271-283. doi: 10.1155/S1073792897000202.

[1]

Woocheol Choi, Youngwoo Koh. On the splitting method for the nonlinear Schrödinger equation with initial data in $ H^1 $. Discrete and Continuous Dynamical Systems, 2021, 41 (8) : 3837-3867. doi: 10.3934/dcds.2021019

[2]

Manh Hong Duong, Yulong Lu. An operator splitting scheme for the fractional kinetic Fokker-Planck equation. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5707-5727. doi: 10.3934/dcds.2019250

[3]

Lassaad Aloui, Imen El Khal El Taief. The Kato smoothing effect for the nonlinear regularized Schrödinger equation on compact manifolds. Mathematical Control and Related Fields, 2020, 10 (4) : 699-714. doi: 10.3934/mcrf.2020016

[4]

Daiwen Huang, Jingjun Zhang. Global smooth solutions for the nonlinear Schrödinger equation with magnetic effect. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 1753-1773. doi: 10.3934/dcdss.2016073

[5]

Jean-Claude Saut, Jun-Ichi Segata. Asymptotic behavior in time of solution to the nonlinear Schrödinger equation with higher order anisotropic dispersion. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 219-239. doi: 10.3934/dcds.2019009

[6]

Guy V. Norton, Robert D. Purrington. The Westervelt equation with a causal propagation operator coupled to the bioheat equation.. Evolution Equations and Control Theory, 2016, 5 (3) : 449-461. doi: 10.3934/eect.2016013

[7]

D.G. deFigueiredo, Yanheng Ding. Solutions of a nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2002, 8 (3) : 563-584. doi: 10.3934/dcds.2002.8.563

[8]

Jibin Li, Yan Zhou. Bifurcations and exact traveling wave solutions for the nonlinear Schrödinger equation with fourth-order dispersion and dual power law nonlinearity. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3083-3097. doi: 10.3934/dcdss.2020113

[9]

Yuto Miyatake, Tai Nakagawa, Tomohiro Sogabe, Shao-Liang Zhang. A structure-preserving Fourier pseudo-spectral linearly implicit scheme for the space-fractional nonlinear Schrödinger equation. Journal of Computational Dynamics, 2019, 6 (2) : 361-383. doi: 10.3934/jcd.2019018

[10]

Pavel I. Naumkin, Isahi Sánchez-Suárez. On the critical nongauge invariant nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2011, 30 (3) : 807-834. doi: 10.3934/dcds.2011.30.807

[11]

Tarek Saanouni. Remarks on the damped nonlinear Schrödinger equation. Evolution Equations and Control Theory, 2020, 9 (3) : 721-732. doi: 10.3934/eect.2020030

[12]

Younghun Hong. Scattering for a nonlinear Schrödinger equation with a potential. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1571-1601. doi: 10.3934/cpaa.2016003

[13]

Alexander Komech, Elena Kopylova, David Stuart. On asymptotic stability of solitons in a nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1063-1079. doi: 10.3934/cpaa.2012.11.1063

[14]

Dario Bambusi, A. Carati, A. Ponno. The nonlinear Schrödinger equation as a resonant normal form. Discrete and Continuous Dynamical Systems - B, 2002, 2 (1) : 109-128. doi: 10.3934/dcdsb.2002.2.109

[15]

Hongwei Wang, Amin Esfahani. On the Cauchy problem for a nonlocal nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022039

[16]

Andreas C. Aristotelous, Ohannes Karakashian, Steven M. Wise. A mixed discontinuous Galerkin, convex splitting scheme for a modified Cahn-Hilliard equation and an efficient nonlinear multigrid solver. Discrete and Continuous Dynamical Systems - B, 2013, 18 (9) : 2211-2238. doi: 10.3934/dcdsb.2013.18.2211

[17]

Mohamad Darwich. On the $L^2$-critical nonlinear Schrödinger Equation with a nonlinear damping. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2377-2394. doi: 10.3934/cpaa.2014.13.2377

[18]

Wulong Liu, Guowei Dai. Multiple solutions for a fractional nonlinear Schrödinger equation with local potential. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2105-2123. doi: 10.3934/cpaa.2017104

[19]

Xudong Shang, Jihui Zhang. Multiplicity and concentration of positive solutions for fractional nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2239-2259. doi: 10.3934/cpaa.2018107

[20]

Patricio Felmer, César Torres. Radial symmetry of ground states for a regional fractional Nonlinear Schrödinger Equation. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2395-2406. doi: 10.3934/cpaa.2014.13.2395

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (83)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]