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 & 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.  Google Scholar

[2]

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

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[6]

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

[7]

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

[8]

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

[9]

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

[10]

M. B. Erdoǧan and N. Tzirakis, Talbot effect for the cubic nonlinear Schrödinger equation on the torus,, preprint, (2013).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[22]

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

[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.  Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

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.  Google Scholar

[2]

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

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[6]

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

[7]

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

[8]

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

[9]

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

[10]

M. B. Erdoǧan and N. Tzirakis, Talbot effect for the cubic nonlinear Schrödinger equation on the torus,, preprint, (2013).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[22]

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

[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.  Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

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