June  2019, 11(2): 239-254. doi: 10.3934/jgm.2019013

Dispersive Lamb systems

School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA

* Corresponding author: Peter J. Olver

Received  October 2017 Revised  March 2018 Published  May 2019

Under periodic boundary conditions, a one-dimensional dispersive medium driven by a Lamb oscillator exhibits a smooth response when the dispersion relation is asymptotically linear or superlinear at large wave numbers, but unusual fractal solution profiles emerge when the dispersion relation is asymptotically sublinear. Strikingly, this is exactly the opposite of the superlinear asymptotic regime required for fractalization and dispersive quantization, also known as the Talbot effect, of the unforced medium induced by discontinuous initial conditions.

Citation: Peter J. Olver, Natalie E. Sheils. Dispersive Lamb systems. Journal of Geometric Mechanics, 2019, 11 (2) : 239-254. doi: 10.3934/jgm.2019013
References:
[1]

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

[2]

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

[3]

V. ChousionisM. B. Erdoğan and N. Tzirakis, Fractal solutions of linear and nonlinear dispersive partial differential equations, Proc. London Math. Soc., 110 (2015), 543-564.  doi: 10.1112/plms/pdu061.

[4]

B. Deconinck, Q. Guo, E. Shlizerman and V. Vasan, Fokas's uniform transform method for linear systems, Quart. Appl. Math., 76 (2018), 463-488, arXiv 1705.00358. doi: 10.1090/qam/1484.

[5]

B. Deconinck, B. Pelloni and N. E. Sheils, Non-steady state heat conduction in composite walls, Proc. Roy. Soc. London A, 470 (2014), 20130605.

[6]

J. J. Duistermaat, Self-similarity of ``Riemann's nondifferentiable function'', Nieuw Arch. Wisk., 9 (1991), 303-337. 

[7]

M. B. Erdoğan, personal communication, 2018.

[8]

M. B. Erdoğan and G. Shakan, Fractal solutions of dispersive partial differential equations on the torus, Selecta Math., 25 (2019), 11. doi: 10.1007/s00029-019-0455-1.

[9]

M. B. Erdoğan and N. Tzirakis, Talbot effect for the cubic nonlinear Schrödinger equation on the torus, Math. Res. Lett., 20 (2013), 1081-1090.  doi: 10.4310/MRL.2013.v20.n6.a7.

[10] M. B. Erdoğan and N. Tzirakis, Dispersive Partial Differential Equations: Wellposedness and Applications, London Math. Soc. Student Texts, vol. 86, Cambridge University Press, Cambridge, 2016.  doi: 10.1017/CBO9781316563267.
[11]

A. S. Fokas, A unified transform method for solving linear and certain nonlinear PDEs, Proc. Roy. Soc. London A, 453 (1997), 1411-1443.  doi: 10.1098/rspa.1997.0077.

[12]

A. S. Fokas, A Unified Approach to Boundary Value Problems, CBMS-NSF Conference Series in Applied Math., vol. 78, SIAM, Philadelphia, 2008. doi: 10.1137/1.9780898717068.

[13]

P. HagertyA. M. Bloch and M. I. Weinstein, Radiation induced instability, Siam J. Appl. Math., 64 (2003), 484-524.  doi: 10.1137/S0036139902418717.

[14]

I. A. Kunin, Elastic Media with Microstructure I, , Springer-Verlag, New York, 1982.

[15]

G. L. Lamb, On a peculiarity of the wave-system due to the free vibrations of a nucleus in an extended medium, Proc. London Math. Soc., 32 (1900), 208-211.  doi: 10.1112/plms/s1-32.1.208.

[16]

P. J. Olver, Dispersive quantization, Amer. Math. Monthly, 117 (2010), 599-610.  doi: 10.4169/000298910x496723.

[17]

P. J. Olver, Introduction to Partial Differential Equations, Undergraduate Texts in Mathematics, Springer, New York, 2014. doi: 10.1007/978-3-319-02099-0.

[18]

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

[19]

A. C. Scott, Soliton oscillations in DNA, Phys. Rev. A, 31 (1985), 3518-3519.  doi: 10.1103/PhysRevA.31.3518.

[20]

N. E. Sheils and B. Deconinck, Heat conduction on the ring: interface problems with periodic boundary conditions, Appl. Math. Lett., 37 (2014), 107-111.  doi: 10.1016/j.aml.2014.06.006.

[21]

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

[22]

H. F. Weinberger, A First Course in Partial Differential Equations, Dover Publ., New York, 1995.

[23]

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

show all references

References:
[1]

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

[2]

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

[3]

V. ChousionisM. B. Erdoğan and N. Tzirakis, Fractal solutions of linear and nonlinear dispersive partial differential equations, Proc. London Math. Soc., 110 (2015), 543-564.  doi: 10.1112/plms/pdu061.

[4]

B. Deconinck, Q. Guo, E. Shlizerman and V. Vasan, Fokas's uniform transform method for linear systems, Quart. Appl. Math., 76 (2018), 463-488, arXiv 1705.00358. doi: 10.1090/qam/1484.

[5]

B. Deconinck, B. Pelloni and N. E. Sheils, Non-steady state heat conduction in composite walls, Proc. Roy. Soc. London A, 470 (2014), 20130605.

[6]

J. J. Duistermaat, Self-similarity of ``Riemann's nondifferentiable function'', Nieuw Arch. Wisk., 9 (1991), 303-337. 

[7]

M. B. Erdoğan, personal communication, 2018.

[8]

M. B. Erdoğan and G. Shakan, Fractal solutions of dispersive partial differential equations on the torus, Selecta Math., 25 (2019), 11. doi: 10.1007/s00029-019-0455-1.

[9]

M. B. Erdoğan and N. Tzirakis, Talbot effect for the cubic nonlinear Schrödinger equation on the torus, Math. Res. Lett., 20 (2013), 1081-1090.  doi: 10.4310/MRL.2013.v20.n6.a7.

[10] M. B. Erdoğan and N. Tzirakis, Dispersive Partial Differential Equations: Wellposedness and Applications, London Math. Soc. Student Texts, vol. 86, Cambridge University Press, Cambridge, 2016.  doi: 10.1017/CBO9781316563267.
[11]

A. S. Fokas, A unified transform method for solving linear and certain nonlinear PDEs, Proc. Roy. Soc. London A, 453 (1997), 1411-1443.  doi: 10.1098/rspa.1997.0077.

[12]

A. S. Fokas, A Unified Approach to Boundary Value Problems, CBMS-NSF Conference Series in Applied Math., vol. 78, SIAM, Philadelphia, 2008. doi: 10.1137/1.9780898717068.

[13]

P. HagertyA. M. Bloch and M. I. Weinstein, Radiation induced instability, Siam J. Appl. Math., 64 (2003), 484-524.  doi: 10.1137/S0036139902418717.

[14]

I. A. Kunin, Elastic Media with Microstructure I, , Springer-Verlag, New York, 1982.

[15]

G. L. Lamb, On a peculiarity of the wave-system due to the free vibrations of a nucleus in an extended medium, Proc. London Math. Soc., 32 (1900), 208-211.  doi: 10.1112/plms/s1-32.1.208.

[16]

P. J. Olver, Dispersive quantization, Amer. Math. Monthly, 117 (2010), 599-610.  doi: 10.4169/000298910x496723.

[17]

P. J. Olver, Introduction to Partial Differential Equations, Undergraduate Texts in Mathematics, Springer, New York, 2014. doi: 10.1007/978-3-319-02099-0.

[18]

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

[19]

A. C. Scott, Soliton oscillations in DNA, Phys. Rev. A, 31 (1985), 3518-3519.  doi: 10.1103/PhysRevA.31.3518.

[20]

N. E. Sheils and B. Deconinck, Heat conduction on the ring: interface problems with periodic boundary conditions, Appl. Math. Lett., 37 (2014), 107-111.  doi: 10.1016/j.aml.2014.06.006.

[21]

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

[22]

H. F. Weinberger, A First Course in Partial Differential Equations, Dover Publ., New York, 1995.

[23]

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

Figure 1.  The Lamb Oscillator on the Line.
Figure 2.  The Lamb Oscillator on the Line at Large Time.
Figure 3.  The Periodic Lamb Oscillator.
Figure 4.  The Dispersive Periodic Lamb Oscillator with $ \omega (k) = k^2 $.
Figure 5.  The Dispersive Periodic Lamb Oscillator for the Klein-Gordon Model.
Figure 6.  The Dispersive Periodic Lamb Oscillator with $ \omega(k) = \sqrt{\left| k \right|} $.
Figure 7.  The Dispersive Periodic Lamb Oscillator for the Regularized Boussinesq Model.
Figure 8.  The Unidirectional Periodic Lamb Oscillator for the Transport Model.
Figure 9.  The Unidirectional Dispersive Periodic Lamb Oscillator for $ \omega(k) = {k^2} $.
Figure 10.  The Unidirectional Dispersive Periodic Lamb Oscillator for $ \omega(k) = \sqrt{k} $.
Figure 11.  The Unidirectional Dispersive Periodic Lamb Oscillator for $ \omega(k)=k^{2} /\left(1+\frac{1}{3} k^{2}\right) $
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