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October  2011, 29(4): 1637-1649. doi: 10.3934/dcds.2011.29.1637

Numerical simulation of resonant tunneling of fast solitons for the nonlinear Schrödinger equation

Walid K. Abou Salem 1, , Xiao Liu 2, and  Catherine Sulem 2,

1. 

Department of Mathematics, University of British Columbia, Vancouver, BC, Canada
2. 

Department of Mathematics, University of Toronto, Toronto, ON, M5S 2E4, Canada, Canada

Received  December 2009 Revised  September 2010 Published  December 2010

In a recent paper [4], we showed that the phenomenon of resonant tunneling, well known in linear quantum mechanical scattering theory, takes place for fast solitons of the Nonlinear Schrödinger (NLS) equation in the presence of certain large potentials. Here, we illustrate numerically this situation for the one dimensional cubic NLS equation with two classes of potentials, namely the 'box' potential and a repulsive 2-delta potential. In particular, under the resonant condition, we show that the transmitted wave is close to a soliton, calculate the transmitted mass of the solution and show that it converges to the total mass of the solution as the velocity of the soliton is increased.
Keywords: Nonlinear Schrödinger equation, Soliton dynamics, resonant tunneling, numerical simulations..
Mathematics Subject Classification: Primary: 35Q55, 37K40; Secondary: 35Q51, 65M6.
Citation: Walid K. Abou Salem, Xiao Liu, Catherine Sulem. Numerical simulation of resonant tunneling of fast solitons for the nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2011, 29 (4) : 1637-1649. doi: 10.3934/dcds.2011.29.1637
References:
[1]

W. K. Abou Salem, Solitary wave dynamics in time-dependent potentials, J. Math. Phys., 49 (2008), 032101. doi: 10.1063/1.2837429.  Google Scholar

[2]

W. K. Abou Salem, Effective dynamics of solitons in the presence of rough nonlinear perturbations, Nonlinearity, 22 (2009), 747-763. doi: 10.1088/0951-7715/22/4/004.  Google Scholar

[3]

W. K. Abou Salem, J. Fröhlich and I. M. Sigal, Colliding solitons for the nonlinear Schrödinger equation, Commun. Math. Phys., 291 (2009), 151-176. doi: 10.1007/s00220-009-0871-8.  Google Scholar

[4]

W. K. Abou Salem and C. Sulem, Resonant tunneling of fast solitons through large potential barriers, to appear in Canad. J. Math., (2010). Google Scholar

[5]

G. D. Akrivis, V. A. Dougalis and O. A. Karakashian, On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation, Numerische Mathematik, 59 (1991), 31-53. doi: 10.1007/BF01385769.  Google Scholar

[6]

G. D. Akrivis, V. A. Dougalis, O. A. Karakashian and W. R. McKinney, Numerical approximation of blow-up of radially symmetric solutions of the nonlinear Schrödinger equation, SIAM J. Sci. Comput., 25 (2003), 186-212. doi: 10.1137/S1064827597332041.  Google Scholar

[7]

J. C. Bronski and R. L. Jerrard, Soliton dynamics in a potential, Math. Res. Lett., 7 (2000), 329-342.  Google Scholar

[8]

J. Fröhlich, S. Gustafson, B. L. G. Jonsson and I. M. Sigal, Solitary wave dynamics in an external potential, Commun. Math. Phys., 250 (2004), 613-642. doi: 10.1007/s00220-004-1128-1.  Google Scholar

[9]

J. Holmer and M. Zworski, Slow soliton interaction with delta impurities, J. Modern Dynamics, 1 (2007), 689-718.  Google Scholar

[10]

J. Holmer and M. Zworski, Soliton interaction with slowly varying potentials, IMRN, (2008), ArtID 026, 36pp.  Google Scholar

[11]

J. Holmer, J. Marzuola and M. Zworski, Fast soliton scattering by delta impurities, Commun. Math. Phys., 274 (2007), 187-216. doi: 10.1007/s00220-007-0261-z.  Google Scholar

[12]

J. Holmer, J. Marzuola and M. Zworski, Soliton splitting by external delta potentials, Journal of Nonlinear Science, 17 (2007), 349-367. doi: 10.1007/s00332-006-0807-9.  Google Scholar

show all references

References:
[1]

W. K. Abou Salem, Solitary wave dynamics in time-dependent potentials, J. Math. Phys., 49 (2008), 032101. doi: 10.1063/1.2837429.  Google Scholar

[2]

W. K. Abou Salem, Effective dynamics of solitons in the presence of rough nonlinear perturbations, Nonlinearity, 22 (2009), 747-763. doi: 10.1088/0951-7715/22/4/004.  Google Scholar

[3]

W. K. Abou Salem, J. Fröhlich and I. M. Sigal, Colliding solitons for the nonlinear Schrödinger equation, Commun. Math. Phys., 291 (2009), 151-176. doi: 10.1007/s00220-009-0871-8.  Google Scholar

[4]

W. K. Abou Salem and C. Sulem, Resonant tunneling of fast solitons through large potential barriers, to appear in Canad. J. Math., (2010). Google Scholar

[5]

G. D. Akrivis, V. A. Dougalis and O. A. Karakashian, On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation, Numerische Mathematik, 59 (1991), 31-53. doi: 10.1007/BF01385769.  Google Scholar

[6]

G. D. Akrivis, V. A. Dougalis, O. A. Karakashian and W. R. McKinney, Numerical approximation of blow-up of radially symmetric solutions of the nonlinear Schrödinger equation, SIAM J. Sci. Comput., 25 (2003), 186-212. doi: 10.1137/S1064827597332041.  Google Scholar

[7]

J. C. Bronski and R. L. Jerrard, Soliton dynamics in a potential, Math. Res. Lett., 7 (2000), 329-342.  Google Scholar

[8]

J. Fröhlich, S. Gustafson, B. L. G. Jonsson and I. M. Sigal, Solitary wave dynamics in an external potential, Commun. Math. Phys., 250 (2004), 613-642. doi: 10.1007/s00220-004-1128-1.  Google Scholar

[9]

J. Holmer and M. Zworski, Slow soliton interaction with delta impurities, J. Modern Dynamics, 1 (2007), 689-718.  Google Scholar

[10]

J. Holmer and M. Zworski, Soliton interaction with slowly varying potentials, IMRN, (2008), ArtID 026, 36pp.  Google Scholar

[11]

J. Holmer, J. Marzuola and M. Zworski, Fast soliton scattering by delta impurities, Commun. Math. Phys., 274 (2007), 187-216. doi: 10.1007/s00220-007-0261-z.  Google Scholar

[12]

J. Holmer, J. Marzuola and M. Zworski, Soliton splitting by external delta potentials, Journal of Nonlinear Science, 17 (2007), 349-367. doi: 10.1007/s00332-006-0807-9.  Google Scholar

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