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 & Continuous Dynamical Systems - A, 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. 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.

[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. doi: 10.1007/BF01896020.

[4]

J. Bourgain, Exponential sums and nonlinear Schrödinger equations,, Geom. Funct. Anal., 3 (1993), 157. 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. doi: 10.1007/BF01895688.

[6]

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

[7]

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

[8]

P. G. Drazin and R. S. Johnson, "Solitons: An Introduction,", Cambridge University Press, (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, (2013).

[11]

M. B. Erdoǧan, N. Tzirakis and V. Zharnitsky, Nearly linear dynamics of nonlinear dispersive waves,, Physica D, 240 (2011), 1325. 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. 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., (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. 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. 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, (2012).

[17]

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

[18]

L. Kapitanski and I. Rodnianski, Does a quantum particle know the time?,, in, 109 (1999), 355. 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. doi: 10.1002/cpa.3160360302.

[20]

C. Lubich, On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations,, Math. Comp., 77 (2008), 2141. 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, (2011).

[22]

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

[23]

K. I. Oskolkov, A class of I.M. Vinogradov's series and its applications in harmonic analysis,, in, 19 (1992), 353. 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. doi: 10.1007/BF02476032.

[25]

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

[26]

J. Smoller, "Shock Waves and Reaction-Diffusion Equations,", 2nd edition, (1994).

[27]

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

[28]

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

[29]

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

[30]

G. B. Whitham, "Linear and Nonlinear Waves,", John Wiley & Sons, (1974).

[31]

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

show all references

References:
[1]

M. V. Berry, Quantum fractals in boxes,, J. Phys. A, 29 (1996), 6617. 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.

[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. doi: 10.1007/BF01896020.

[4]

J. Bourgain, Exponential sums and nonlinear Schrödinger equations,, Geom. Funct. Anal., 3 (1993), 157. 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. doi: 10.1007/BF01895688.

[6]

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

[7]

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

[8]

P. G. Drazin and R. S. Johnson, "Solitons: An Introduction,", Cambridge University Press, (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, (2013).

[11]

M. B. Erdoǧan, N. Tzirakis and V. Zharnitsky, Nearly linear dynamics of nonlinear dispersive waves,, Physica D, 240 (2011), 1325. 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. 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., (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. 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. 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, (2012).

[17]

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

[18]

L. Kapitanski and I. Rodnianski, Does a quantum particle know the time?,, in, 109 (1999), 355. 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. doi: 10.1002/cpa.3160360302.

[20]

C. Lubich, On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations,, Math. Comp., 77 (2008), 2141. 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, (2011).

[22]

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

[23]

K. I. Oskolkov, A class of I.M. Vinogradov's series and its applications in harmonic analysis,, in, 19 (1992), 353. 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. doi: 10.1007/BF02476032.

[25]

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

[26]

J. Smoller, "Shock Waves and Reaction-Diffusion Equations,", 2nd edition, (1994).

[27]

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

[28]

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

[29]

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

[30]

G. B. Whitham, "Linear and Nonlinear Waves,", John Wiley & Sons, (1974).

[31]

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

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