A Hamilton–Jacobi approach to the control of the trapping time of a soliton by an external potential
Authors:
G. Kossioris, M. Plexousakis and A. N. Yannacopoulos
Journal:
Quart. Appl. Math. 63 (2005), 309-324
MSC (2000):
Primary 49L25
DOI:
https://doi.org/10.1090/S0033-569X-05-00950-0
Published electronically:
February 28, 2005
MathSciNet review:
2150777
Full-text PDF Free Access
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Abstract: The control of the trapping time of a localized solution of the nonlinear Schrödinger equation (NLS) with the use of an external parabolic potential is studied. We reduce the dynamics of the position of the soliton center to those of a controlled linear oscillator and then study the viscosity solution of the associated Hamilton-Jacobi equation. A numerical scheme is proposed for the treatment of the problem.
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- A. N. Yannacopoulos, D. J. Frantzeskakis, C. Polymilis, and K. Hizanidis, Conditions for soliton trapping in random potentials using Lyapunov exponents of stochastic ODEs, Phys. Lett. A 271 (2000), no. 5-6, 334–340. MR 1773301, DOI https://doi.org/10.1016/S0375-9601%2800%2900348-0
ANY2 A. N. Yannacopoulos, D. J. Frantzeskakis, C. Polymilis and K. Hizanidis, Motion of 2D Schrödinger Solitary Waves in the Presence of Random External Potentials, Physica Scr. 65 (2002), 363–368.
Zhao H. Zhao, A fast sweeping method for eikonal equations, to appear in Math. Comp.
ABP F. K. Abdullaev, J. C. Bronski and G. Papanicolaou, Soliton perturbations and the random Kepler problem, Physica D 135 (2000), 369–386.
Bardi1 M. Bardi, A boundary value problem for the minimum time function, SIAM J. Control and Optimization 27 (1989), 776–785.
Bardi M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations, Birkhäuser Boston, Inc., Boston, 1997.
Bardi2 M. Bardi and M. Falcone, An approximation scheme for the minimum time function, SIAM J. Control and Optimization 28 (1990), 950–965.
BaP G. Barles and B. Perthame, Comparison Principle for Dirichlet-type Hamilton-Jacobi Equations and Singular Perturbations of Degenerate Elliptic Equations, Appl. Math. Optim. 21 (1990), 21–44.
BaSoug G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Analysis 4 (1991), 271–283.
CIL M. G. Crandall, H. Ishii, and P.-L. Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. 27 (1992), 1–67.
CrLi M. G. Crandall and P.-L. Lions, Two approximations of solutions of Hamilton–Jacobi equations, Math. Comp. 43 (1984), 1–19.
Garnier J. Garnier, Asymptotic transmission of solitons through random media, SIAM J. Appl. Math. 58 (1998), 1969–1995.
HasKod A. Hasegawa, Y. Kodama and Y. Kodama, Solitons in optical communications, Oxford Series in Optical and Imaging Science 7, Clarendon Press, Oxford 1997.
PitStr L. Pitaevskii and S. Stringeri, Bose–Einstein condensation, International Series of Monographs on Physics, Clarendon Press, Oxford 2003.
RouyT E. Rouy and A. Tourin, A viscosity solution approach to shape-from-shading, SIAM J. Num. Anal. 29 (1992), 867–884.
ANY1 A. N. Yannacopoulos, D. J. Frantzeskakis, C. Polymilis and K. Hizanidis, Conditions for soliton trapping in random potentials using Lyapunov exponents of stochastic ODEs, Phys. Lett. A 271 (2000), 334–340.
ANY2 A. N. Yannacopoulos, D. J. Frantzeskakis, C. Polymilis and K. Hizanidis, Motion of 2D Schrödinger Solitary Waves in the Presence of Random External Potentials, Physica Scr. 65 (2002), 363–368.
Zhao H. Zhao, A fast sweeping method for eikonal equations, to appear in Math. Comp.
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Additional Information
G. Kossioris
Affiliation:
Department of Mathematics, University of Crete, 71409 Heraklion, Greece ; Institute of Applied and Computational Mathematics, FORTH, 71110 Heraklion, Greece
Email:
kosioris@math.uoc.gr
M. Plexousakis
Affiliation:
Department of Applied Mathematics, University of Crete, 71409 Heraklion, Greece ; Institute of Applied and Computational Mathematics, FORTH, 71110 Heraklion, Greece
Email:
plex@tem.uoc.gr
A. N. Yannacopoulos
Affiliation:
Department of Statistics and Actuarial Science, University of the Aegean, 82300 Karlovassi, Samos, Greece
Email:
ayannaco@aegean.gr
Received by editor(s):
June 20, 2004
Published electronically:
February 28, 2005
Article copyright:
© Copyright 2005
Brown University
