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

Walid K. Abou Salem; Xiao Liu; Catherine Sulem

doi:10.3934/dcds.2011.29.1637

Article Contents

2011, Volume 29, Issue 4: 1637-1649. Doi: 10.3934/dcds.2011.29.1637

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

1.
Department of Mathematics, University of British Columbia, Vancouver, BC
2.
Department of Mathematics, University of Toronto, Toronto, ON, M5S 2E4

Received: November 30, 2009

Revised: August 31, 2010

Published: September 2011

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Abstract

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.

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Mathematics Subject Classification: Primary: 35Q55, 37K40; Secondary: 35Q51, 65M60.

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References

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