메뉴 건너뛰기




Volumn 36, Issue 1, 2014, Pages

A modulus-squared Dirichlet boundary condition for time-dependent complex partial differential equations and its application to the nonlinear Schroinger equation

Author keywords

Complex partial differential equations; Nonlinear Schrodinger equation; Numerical boundary conditions

Indexed keywords

BOUNDARY CONDITIONS; NONLINEAR EQUATIONS; NUMERICAL MODELS;

EID: 84896974195     PISSN: 10648275     EISSN: 10957200     Source Type: Journal    
DOI: 10.1137/130920046     Document Type: Article
Times cited : (19)

References (46)
  • 1
    • 44349167220 scopus 로고    scopus 로고
    • Vortex rings in toroidal Bose-Einstein condensates
    • M. Abad, M. Guilleumas, R. Mayol, and M. Pi, Vortex rings in toroidal Bose-Einstein condensates, Laser Phys., 18 (2008), pp. 648-652.
    • (2008) Laser Phys. , vol.18 , pp. 648-652
    • Abad, M.1    Guilleumas, M.2    Mayol, R.3    Pi, M.4
  • 2
    • 51149087560 scopus 로고    scopus 로고
    • A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrodinger equations
    • X. Antoine, A. Arnold, C. Besse, M. Ehrhardt, and A. Schadle, A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrodinger equations, Comm. Comput. Phys., 4 (2008), pp. 729-796.
    • (2008) Comm. Comput. Phys. , vol.4 , pp. 729-796
    • Antoine, X.1    Arnold, A.2    Besse, C.3    Ehrhardt, M.4    Schadle, A.5
  • 3
    • 84888301753 scopus 로고    scopus 로고
    • Computational methods for the dynamics of the nonlinear Schrodinger/Gross-Pitaevskii equations
    • X. Antoine, W. Bao, and C. Besse, Computational methods for the dynamics of the nonlinear Schrodinger/Gross-Pitaevskii equations, Comput. Phys. Comm., 184 (2013), pp. 2621-2633.
    • (2013) Comput. Phys. Comm. , vol.184 , pp. 2621-2633
    • Antoine, X.1    Bao, W.2    Besse, C.3
  • 4
    • 33750189006 scopus 로고    scopus 로고
    • Artificial boundary conditions for one-dimensional cubic nonlinear Schrodinger equations
    • X. Antoine, C. Besse, and S. Descombes, Artificial boundary conditions for one-dimensional cubic nonlinear Schrodinger equations, SIAM J. Numer. Anal., 43 (2006), pp. 2272-2293.
    • (2006) SIAM J. Numer. Anal. , vol.43 , pp. 2272-2293
    • Antoine, X.1    Besse, C.2    Descombes, S.3
  • 5
    • 56749091936 scopus 로고    scopus 로고
    • Absorbing boundary conditions for the one-dimensional Schrodinger equation with an exterior repulsive potential
    • X. Antoine, C. Besse, and P. Klein, Absorbing boundary conditions for the one-dimensional Schrodinger equation with an exterior repulsive potential, J. Comput. Phys., 228 (2009), pp. 312-335.
    • (2009) J. Comput. Phys. , vol.228 , pp. 312-335
    • Antoine, X.1    Besse, C.2    Klein, P.3
  • 6
    • 79957606319 scopus 로고    scopus 로고
    • Absorbing boundary conditions for general nonlinear Schrodinger equations
    • X. Antoine, C. Besse, and P. Klein, Absorbing boundary conditions for general nonlinear Schrodinger equations, SIAM J. Sci. Comput., 33 (2011), pp. 1008-1033.
    • (2011) SIAM J. Sci. Comput. , vol.33 , pp. 1008-1033
    • Antoine, X.1    Besse, C.2    Klein, P.3
  • 7
    • 80051600356 scopus 로고    scopus 로고
    • Numerical solution of time-dependent nonlinear Schrodinger equations using domain truncation techniques coupled with relaxation scheme
    • X. Antoine, C. Besse, and P. Klein, Numerical solution of time-dependent nonlinear Schrodinger equations using domain truncation techniques coupled with relaxation scheme, Laser Phys., 21 (2011), pp. 1191-1502.
    • (2011) Laser Phys. , vol.21 , pp. 1191-1502
    • Antoine, X.1    Besse, C.2    Klein, P.3
  • 8
    • 84864820230 scopus 로고    scopus 로고
    • Absorbing boundary conditions for the two-dimensional Schrodinger equation with an exterior potential: Part I: Construction and a priori estimates
    • X. Antoine, C. Besse, and P. Klein, Absorbing boundary conditions for the two-dimensional Schrodinger equation with an exterior potential: Part I: Construction and a priori estimates, Math. Models Methods Appl. Sci., 22 (2012), pp. 50026-50064.
    • (2012) Math. Models Methods Appl. Sci. , vol.22 , pp. 50026-50064
    • Antoine, X.1    Besse, C.2    Klein, P.3
  • 9
    • 84884210870 scopus 로고    scopus 로고
    • Absorbing boundary conditions for the two-dimensional Schroodinger equation with an exterior potential: Part II: Discretization and numerical results
    • X. Antoine, C. Besse, and P. Klein, Absorbing boundary conditions for the two-dimensional Schroodinger equation with an exterior potential: Part II: Discretization and numerical results, Numer. Math., 125 (2013), pp. 191-223.
    • (2013) Numer. Math. , vol.125 , pp. 191-223
    • Antoine, X.1    Besse, C.2    Klein, P.3
  • 10
    • 84872170647 scopus 로고    scopus 로고
    • Mathematical theory and numerical methods for Bose-Einstein condensation
    • W. Bao and Y. Cai, Mathematical theory and numerical methods for Bose-Einstein condensation, Kinetic Related Models, 6 (2013), pp. 1-135.
    • (2013) Kinetic Related Models , vol.6 , pp. 1-135
    • Bao, W.1    Cai, Y.2
  • 11
    • 84871016569 scopus 로고    scopus 로고
    • Numerical methods and comparison for computing dark and bright solitons in the nonlinear Schrodinger equation
    • W. Bao, Q. Tang, and Z. Xu, Numerical methods and comparison for computing dark and bright solitons in the nonlinear Schrodinger equation, J. Comput. Phys., 235 (2013), pp. 423-445.
    • (2013) J. Comput. Phys. , vol.235 , pp. 423-445
    • Bao, W.1    Tang, Q.2    Xu, Z.3
  • 12
    • 28044459877 scopus 로고
    • A perfectly matched layer for the absorption of electromagnetic waves
    • J. P. Bérenger, A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys., 114 (1994), pp. 185-200.
    • (1994) J. Comput. Phys. , vol.114 , pp. 185-200
    • Bérenger, J.P.1
  • 13
    • 0034625209 scopus 로고    scopus 로고
    • Motions in a Bose condensate: VII. Boundary-layer separation
    • N. G. Berloff and P. H. Roberts, Motions in a Bose condensate: VII. Boundary-layer separation, J. Phys. A, 33 (2000), p. 4025.
    • (2000) J. Phys. a , vol.33 , pp. 4025
    • Berloff, N.G.1    Roberts, P.H.2
  • 14
    • 0030101001 scopus 로고    scopus 로고
    • A history of Runge-Kutta methods
    • J. C. Butcher, A history of Runge-Kutta methods, Appl. Numer. Math., 20 (1996), pp. 247-260.
    • (1996) Appl. Numer. Math. , vol.20 , pp. 247-260
    • Butcher, J.C.1
  • 16
    • 84873107900 scopus 로고    scopus 로고
    • NLSEmagic: Nonlinear Schrodinger equation multi-dimensional Matlab-based GPU-accelerated integrators using compact high-order schemes
    • R. M. Caplan, NLSEmagic: Nonlinear Schrodinger equation multi-dimensional Matlab-based GPU-accelerated integrators using compact high-order schemes, Comput. Phys. Comm., 184 (2013), pp. 1250-1271.
    • (2013) Comput. Phys. Comm. , vol.184 , pp. 1250-1271
    • Caplan, R.M.1
  • 17
    • 84896957193 scopus 로고    scopus 로고
    • Numerical stability of explicit Runge-Kutta finite-difference schemes for the nonlinear Schrodinger equation
    • submitted
    • R. M. Caplan and R. Carretero-González, Numerical stability of explicit Runge-Kutta finite-difference schemes for the nonlinear Schrodinger equation, Appl. Num. Math., submitted.
    • Appl. Num. Math.
    • Caplan, R.M.1    Carretero-González, R.2
  • 18
    • 84876273862 scopus 로고    scopus 로고
    • A two-step high order compact scheme for the Laplacian operator and its implementation in a fully explicit method for integrating the nonlinear Schrodinger equation
    • R. M. Caplan and R. Carretero-González, A two-step high order compact scheme for the Laplacian operator and its implementation in a fully explicit method for integrating the nonlinear Schrodinger equation, J. Comput. Appl. Math., 251 (2013), pp. 33-46.
    • (2013) J. Comput. Appl. Math. , vol.251 , pp. 33-46
    • Caplan, R.M.1    Carretero-González, R.2
  • 19
    • 0002924728 scopus 로고
    • Stable and accurate boundary treatments for compact, high-order finite-difference schemes
    • M. H. Carpenter, D. Gottlieb, and S. Abarbanel, Stable and accurate boundary treatments for compact, high-order finite-difference schemes, Appl. Numer. Math., 12 (1993), pp. 55-87.
    • (1993) Appl. Numer. Math. , vol.12 , pp. 55-87
    • Carpenter, M.H.1    Gottlieb, D.2    Abarbanel, S.3
  • 20
    • 0001473098 scopus 로고
    • Time-stable boundary conditions for finite-difference schemes solving hyperbolic systems: Methodology and application to highorder compact schemes
    • M. H. Carpenter, D. Gottlieb, and S. Abarbanel, Time-stable boundary conditions for finite-difference schemes solving hyperbolic systems: Methodology and application to highorder compact schemes, J. Comput. Phys., 111 (1994), pp. 220-236.
    • (1994) J. Comput. Phys. , vol.111 , pp. 220-236
    • Carpenter, M.H.1    Gottlieb, D.2    Abarbanel, S.3
  • 21
    • 45149099903 scopus 로고    scopus 로고
    • Nonlinear waves in Bose-Einstein condensates: Physical relevance and mathematical techniques
    • R. Carretero-González, D. J. Frantzeskakis, and P. G. Kevrekidis, Nonlinear waves in Bose-Einstein condensates: Physical relevance and mathematical techniques, Nonlinearity, 21 (2008), pp. R139-R202.
    • (2008) Nonlinearity , vol.21
    • Carretero-González, R.1    Frantzeskakis, D.J.2    Kevrekidis, P.G.3
  • 24
    • 0013027309 scopus 로고
    • Quantized vortices in helium II
    • Cambridge University Press, Cambridge
    • R. J. Donnelly, Quantized Vortices in Helium II, Cambridge Stud. Low Temp. Phys. 3, Cambridge University Press, Cambridge, 1991.
    • (1991) Cambridge Stud. Low Temp. Phys. 3
    • Donnelly, R.J.1
  • 25
    • 84859640337 scopus 로고    scopus 로고
    • Open boundary conditions: Fact and fiction, in IUTAM Symposium on Advances in Mathematical Modelling of Atmosphere and Ocean Dynamics
    • P. F. Hodnett, ed., Springer, Dordrecht
    • D. R. Durran, Open boundary conditions: Fact and fiction, in IUTAM Symposium on Advances in Mathematical Modelling of Atmosphere and Ocean Dynamics, P. F. Hodnett, ed., Fluid Mech. Appl. 61, Springer, Dordrecht, 2001, pp. 1-18.
    • (2001) Fluid Mech. Appl. 61 , pp. 1-18
    • Durran, D.R.1
  • 27
    • 0036577070 scopus 로고    scopus 로고
    • Generating vortex rings in Bose-Einstein condensates in the line-source approximation
    • M. Guilleumas, D. M. Jezeka, R. Mayol, M. Pi, and M. Barranco, Generating vortex rings in Bose-Einstein condensates in the line-source approximation, Phys. Rev. A (3), 65 (2002), 053609.
    • (2002) Phys. Rev. A (3) , vol.65 , pp. 053609
    • Guilleumas, M.1    Jezeka, D.M.2    Mayol, R.3    Pi, M.4    Barranco, M.5
  • 28
    • 33750158742 scopus 로고    scopus 로고
    • Bending-wave instability of a vortex ring in a trapped Bose-Einstein condensate
    • T. L. Horng, S. C. Gou, and T. C. Lin, Bending-wave instability of a vortex ring in a trapped Bose-Einstein condensate, Phys. Rev. A (3), 74 (2006), 041603(R).
    • (2006) Phys. Rev. A (3) , vol.74
    • Horng, T.L.1    Gou, S.C.2    Lin, T.C.3
  • 29
    • 46349107240 scopus 로고    scopus 로고
    • Transition to quantum turbulence in a Bose-Einstein condensate through the bending-wave instability of a single-vortex ring
    • T.-L. Horng, C.-H. Hsueh, and S.-C. Gou, Transition to quantum turbulence in a Bose-Einstein condensate through the bending-wave instability of a single-vortex ring, Phys. Rev. A (3), 77 (2008), 063625.
    • (2008) Phys. Rev. A (3) , vol.77 , pp. 063625
    • Horng, T.-L.1    Hsueh, C.-H.2    Gou, S.-C.3
  • 30
    • 36348962375 scopus 로고    scopus 로고
    • Vortex-ring solutions of the Gross-Pitaevskii equation for an axisymmtrically trapped Bose-Einstein condensate
    • C. H. Hsueh, S. C. Gou, T. L. Horng, and Y. M. Kao, Vortex-ring solutions of the Gross-Pitaevskii equation for an axisymmtrically trapped Bose-Einstein condensate, J. Phys. B, 40 (2007), pp. 4561-4571.
    • (2007) J. Phys. B , vol.40 , pp. 4561-4571
    • Hsueh, C.H.1    Gou, S.C.2    Horng, T.L.3    Kao, Y.M.4
  • 31
    • 18844480787 scopus 로고    scopus 로고
    • Vortex line and ring dynamics in trapped Bose-Einstein condensates
    • B. Jackson, J. F. McCann, and C. S. Adams, Vortex line and ring dynamics in trapped Bose-Einstein condensates, Phys. Rev. A (3), 61 (1999), 013604.
    • (1999) Phys. Rev. A (3) , vol.61 , pp. 013604
    • Jackson, B.1    McCann, J.F.2    Adams, C.S.3
  • 32
    • 3242678233 scopus 로고    scopus 로고
    • Solving Nonlinear Equations with Newton's Method
    • SIAM, Philadelphia
    • C. T. Kelley, Solving Nonlinear Equations with Newton's Method, Fundam. Algorithms, SIAM, Philadelphia, 2003.
    • (2003) Fundam. Algorithms
    • Kelley, C.T.1
  • 35
    • 34548303395 scopus 로고    scopus 로고
    • Vortex rings and solitary waves in trapped Bose-Einstein condensates
    • S. Komineas, Vortex rings and solitary waves in trapped Bose-Einstein condensates, Eur. Phys. J. Spec. Top., 147 (2007), pp. 133-152.
    • (2007) Eur. Phys. J. Spec. Top. , vol.147 , pp. 133-152
    • Komineas, S.1
  • 36
    • 27144522007 scopus 로고    scopus 로고
    • Collisions of solitons and vortex rings in cylindrical Bose-Einstein condensates
    • S. Komineas and J. Brand, Collisions of solitons and vortex rings in cylindrical Bose-Einstein condensates, Phys. Rev. Lett, 95 (2005), 110401.
    • (2005) Phys. Rev. Lett , vol.95 , pp. 110401
    • Komineas, S.1    Brand, J.2
  • 37
    • 77956705971 scopus 로고    scopus 로고
    • Variational methods in nonlinear fiber-optics and related fields
    • B. A. Malomed, Variational methods in nonlinear fiber-optics and related fields, Progr. Opt., 43 (2002), pp. 71-193.
    • (2002) Progr. Opt. , vol.43 , pp. 71-193
    • Malomed, B.A.1
  • 38
    • 61849133372 scopus 로고    scopus 로고
    • On the convergence of splitting methods for linear evolutionary Schrodinger equations involving an unbounded potential
    • C. Neuhauser and M. Thalhammer, On the convergence of splitting methods for linear evolutionary Schrodinger equations involving an unbounded potential, BIT, 49 (2009), pp. 199-215.
    • (2009) BIT , vol.49 , pp. 199-215
    • Neuhauser, C.1    Thalhammer, M.2
  • 39
    • 0021638932 scopus 로고
    • A radiation boundary condition for multi-dimensional flows
    • W. H. Raymond and H. L. Kuo, A radiation boundary condition for multi-dimensional flows, Quart. J. Roy. Meteor. Soc., 110 (1984), pp. 535-551.
    • (1984) Quart. J. Roy. Meteor. Soc. , vol.110 , pp. 535-551
    • Raymond, W.H.1    Kuo, H.L.2
  • 40
    • 0007082224 scopus 로고
    • Motions in a Bose condensate I. The structure of the large circular vortex
    • P. H. Roberts and J. Grant, Motions in a Bose condensate I. The structure of the large circular vortex, J. Phys. A, 4 (1971), pp. 55-72.
    • (1971) J. Phys. a , vol.4 , pp. 55-72
    • Roberts, P.H.1    Grant, J.2
  • 41
    • 84860423498 scopus 로고    scopus 로고
    • Crow instability in trapped Bose-Einstein condensates
    • T. P. Simula, Crow instability in trapped Bose-Einstein condensates, Phys. Rev. A (3), 84 (2011), 021603(R).
    • (2011) Phys. Rev. A (3) , vol.84
    • Simula, T.P.1
  • 43
    • 0003230098 scopus 로고    scopus 로고
    • The nonlinear schrodinger equation, self-focusing and wave collapse
    • Springer, New York
    • C. Sulem and P. L. Sulem, The Nonlinear Schrodinger Equation, Self-Focusing and Wave Collapse, Appl. Math. Sci. 139, Springer, New York, 1999.
    • (1999) Appl. Math. Sci. 139
    • Sulem, C.1    Sulem, P.L.2
  • 44
    • 38049061810 scopus 로고    scopus 로고
    • The dynamics and interaction of quantized vortices in the Ginzburg-Landau-Schrodinger equation
    • Y. Zhang, W. Bao, and Q. Du, The dynamics and interaction of quantized vortices in the Ginzburg-Landau-Schrodinger equation, SIAM J. Appl. Math., 67 (2007), pp. 1740-1775.
    • (2007) SIAM J. Appl. Math. , vol.67 , pp. 1740-1775
    • Zhang, Y.1    Bao, W.2    Du, Q.3
  • 45
    • 35148816070 scopus 로고    scopus 로고
    • Numerical simulation of vortex dynamics in Ginzburg-Landau-Schrodinger equation
    • Y. Zhang, W. Bao, and Q. Du, Numerical simulation of vortex dynamics in Ginzburg-Landau-Schrodinger equation, Eur. J. Appl. Math., 18 (2007), pp. 607-630.
    • (2007) Eur. J. Appl. Math. , vol.18 , pp. 607-630
    • Zhang, Y.1    Bao, W.2    Du, Q.3
  • 46
    • 35348884878 scopus 로고    scopus 로고
    • A perfectly matched layer approach to the nonlinear Schrodinger wave equations
    • C. Zheng, A perfectly matched layer approach to the nonlinear Schrodinger wave equations, J. Comput. Phys., 227 (2007), pp. 537-556.
    • (2007) J. Comput. Phys. , vol.227 , pp. 537-556
    • Zheng, C.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.