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Numerical methods for a class of generalized nonlinear Schrödinger equations
1. | Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, 10117 Berlin, Germany, Germany |
2. | Vilnius Gediminas Technical University, Saulėtekio al. 11, LT-10223 Vilnius, Lithuania |
References:
[1] |
M. J. Ablowitz and T. P. Horikis, Solitons and spectral renormalization methods in nonlinear optics, Eur. Phys. J. Special Topics, 173 (2009), 147-166.
doi: 10.1140/epjst/e2009-01072-0. |
[2] |
G. P. Agrawal, Nonlinear Fiber Optics, $4^{th}$ edition, Academic, New York, 2007. |
[3] |
N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams, Chapman and Hall, 1997. |
[4] |
N. Akhmediev and M. Karlsson, Cherenkov radiation emitted by solitons in optical fibers, Phys. Rev. A, 51 (1995), 2602-2607.
doi: 10.1103/PhysRevA.51.2602. |
[5] |
G. D. Akrivis and V. D. Dougalis, On a class of conservative, highly accurate Galerkin methods for the Schrödinger equation, RAIRO Model. Math. Anal. Numer., 25 (1991), 643-670. |
[6] |
G. D. Akrivis, V. D. Dougalis and V. A. Karakashian, On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation, Numer. Math., 59 (1991), 31-53.
doi: 10.1007/BF01385769. |
[7] |
Sh. Amiranashvili, U. Bandelow and A. Mielke, Padé approximant for refractive index and nonlocal envelope equations, Opt. Comm., 283 (2010), 480-485.
doi: 10.1016/j.optcom.2009.10.034. |
[8] |
Sh. Amiranashvili and A. Demircan, Hamiltonian structure of propagation equations for ultrashort optical pulses, Phys. Rev. A, 82 (2010), 013812.
doi: 10.1103/PhysRevA.82.013812. |
[9] |
Sh. Amiranashvili and A. Demircan, Ultrashort optical pulse propagation in terms of analytic signal, Advances in Optical Technologies, 2011 (2011), 989515, 8pp.
doi: 10.1155/2011/989515. |
[10] |
X. Antoine, A. Arnold, Ch. Besse, M. Ehrhardt and A. Schädle, A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations, Commun. Comput. Physics, 4 (2008), 729-796. |
[11] |
X. Antoine, W. Bao and C. Besse, Computational methods for the dynamics of nonlinear schrödinger/gross-pitaevskii equations, Comput. Phys. Commun., 184 (2013), 2621-2633.
doi: 10.1016/j.cpc.2013.07.012. |
[12] |
W. Bao, Sh. Jin and P. A. Markowich, Numerical study of time-splitting spectral discretizations of nonlinear Schrödinger equations in the semiclassical regimes, SIAM J. Sci. Comput., 25 (2003), 27-64.
doi: 10.1137/S1064827501393253. |
[13] |
W. Bao, Q. Tang and Z. Xu, Numerical methods and comparison for computing dark and bright solitons in the nonlinear schrödinger equation, J. Comput. Phys., 235 (2013), 423-445.
doi: 10.1016/j.jcp.2012.10.054. |
[14] |
M. Bass, E. W. Van Stryland, D. R. Williams and W. L. Wolfe, editors, Handbook of Optics, Volume 1, $2^{nd}$ edition, McGraw-Hill, 1995. |
[15] |
A. Borzi and E. Decker, Analysis of a leap-frog pseudospectral scheme for the Schrödinger equation, J. Comput. Appl. Math., 193 (2006), 65-88.
doi: 10.1016/j.cam.2005.04.066. |
[16] |
R. W. Boyd, Nonlinear Optics, $3^rd$ edition, Academic, New York, 2008. |
[17] |
Q. Chang, E. Jia and W. Sun, Difference schemes for solving the generalized nonlinear schrödinger equation, J. Comput. Phys., 148 (1999), 397-415.
doi: 10.1006/jcph.1998.6120. |
[18] |
R. Čiegis, I. Laukaitytė and M. Radziunas, Numerical algorithms for Schrödinger equations with artificial boundary conditions, Numer. Funct. Anal. Optim., 30 (2009), 903-923.
doi: 10.1080/01630560903393097. |
[19] |
R. Čiegis and M. Radziunas, Effective numerical integration of traveling wave model for edge-emitting broad-area semiconductor lasers and amplifiers, Math. Model. Anal., 15 (2010), 409-430.
doi: 10.3846/1392-6292.2010.15.409-430. |
[20] |
A. Demircan, Sh. Amiranashvili, C. Brée and G. Steinmeyer, Compressible octave spanning supercontinuum generation by two-pulse collisions, Phys. Rev. Lett., 110 (2013), 233901.
doi: 10.1103/PhysRevLett.110.233901. |
[21] |
A. Demircan, Sh. Amiranashvili and G. Steinmeyer, Controlling light by light with an optical event horizon, Phys. Rev. Lett., 106 (2011), 163901.
doi: 10.1103/PhysRevLett.106.163901. |
[22] |
J. M. Dudley, G. Genty and S. Coen, Supercontinuum generation in photonic crystal fiber, Rev. Mod. Phys., 78 (2006), 1135-1184.
doi: 10.1103/RevModPhys.78.1135. |
[23] |
M. D. Feit, J. A. Fleck and A. Steiger, Solution of the Schrödinger equation by a spectral method, J. Comput. Phys., 47 (1982), 412-433.
doi: 10.1016/0021-9991(82)90091-2. |
[24] | |
[25] |
W. Hundsdorfer and J. G. Verwer, Numerical Solution of Time-Dependent Advection-Difusion-Reaction Equations, Springer Series in Computational Mathematics, 33, Springer, Berlin, Heidelberg, New York, Tokyo, 2003.
doi: 10.1007/978-3-662-09017-6. |
[26] |
A. J. Lichtenberg and M. A. Lieberman, Regular and Chaotic Dynamics, Applied Mathematical Sciences, 38, $2^{nd}$ edition, Springer, 1992.
doi: 10.1007/978-1-4757-2184-3. |
[27] |
C. Lubich, On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations, Math. Comput., 77 (2008), 2141-2153.
doi: 10.1090/S0025-5718-08-02101-7. |
[28] |
G. Muslu and H. Erbay, Higher-order split-step Fourier schemes for the generalized nonlinear Schrödinger equation, Mathematics and Computers in Simulation, 67 (2005), 581-595.
doi: 10.1016/j.matcom.2004.08.002. |
[29] |
Ch. Neuhauser and M. Thalhammer, On the convergence of splitting methods for linear evolutionary Schrödinger equations involving an unbounded potential, BIT Numer. Math., 49 (2009), 199-215.
doi: 10.1007/s10543-009-0215-2. |
[30] |
D. Pathria and J. L. Morris, Exact solutions for a generalized nonlinear Schrödinger equation, Phys. Scripta, 39 (1989), 673-680.
doi: 10.1088/0031-8949/39/6/001. |
[31] |
D. Pathria and J. L. Morris, Pseudo-spectral solution of nonlinear Schrödinger equations, J. Comput. Phys., 87 (1990), 108-125.
doi: 10.1016/0021-9991(90)90228-S. |
[32] |
M. Radziunas, R. Čiegis and A. Mirinavičius, On compact high order finite difference schemes for linear Schrödinger problem on non-uniform meshes, Int. Journal of Num. Anal. Model., 11 (2014), 303-314. |
[33] |
J. M. Sanz-Serna and J. G. Verwer, Conservative and non-conservative schemes for the solution of the nonlinear Schrödinger equation, IMA J. Numer. Anal., 6 (1986), 25-42.
doi: 10.1093/imanum/6.1.25. |
[34] |
J. M. Sanz-Serna and M. P. Calvo, Numerical Hamiltonian Problems, Applied Mathematics and Mathematical Computation, 7, Chapman & Hall, London, 1994. |
[35] |
J. M. Stone and J. C. Knight, Visibly white light generation in uniform photonic crystal fiber using a microchip laser, Opt. Express, 16 (2008), 2670-2675.
doi: 10.1364/OE.16.002670. |
[36] |
G. Strang, On the construction and comparison of difference schemes, SIAM J. Numer. Anal., 5 (1968), 506-517.
doi: 10.1137/0705041. |
[37] |
T. R. Taha and M. J. Ablovitz, Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical Schrödinger equation, J. Comp. Phys., 55 (1984), 203-230.
doi: 10.1016/0021-9991(84)90003-2. |
[38] |
M. Thalhammer, M. Caliari and Ch. Neuhauser, High-order time-splitting Hermite and Fourier spectral methods, J. Comput. Phys., 228 (2009), 822-832.
doi: 10.1016/j.jcp.2008.10.008. |
[39] |
D. E. Vakman and L. A. Vainshtein, Amplitude, phase, frequency - fundamental concepts of oscillation theory, Usp. Fiz. Nauk, 20 (1977), 1002-1016.
doi: 10.1070/PU1977v020n12ABEH005479. |
[40] |
P. K. A. Wai, H. H. Chen and Y. C. Lee, Radiations by "solitons" at the zero group-dispersion wavelength of single-mode optical fibers, Phys. Rev. A, 41 (1990), 426-439.
doi: 10.1103/PhysRevA.41.426. |
[41] |
V. E. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Sov. Phys. JETP, 34 (1972), 62-69. |
show all references
References:
[1] |
M. J. Ablowitz and T. P. Horikis, Solitons and spectral renormalization methods in nonlinear optics, Eur. Phys. J. Special Topics, 173 (2009), 147-166.
doi: 10.1140/epjst/e2009-01072-0. |
[2] |
G. P. Agrawal, Nonlinear Fiber Optics, $4^{th}$ edition, Academic, New York, 2007. |
[3] |
N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams, Chapman and Hall, 1997. |
[4] |
N. Akhmediev and M. Karlsson, Cherenkov radiation emitted by solitons in optical fibers, Phys. Rev. A, 51 (1995), 2602-2607.
doi: 10.1103/PhysRevA.51.2602. |
[5] |
G. D. Akrivis and V. D. Dougalis, On a class of conservative, highly accurate Galerkin methods for the Schrödinger equation, RAIRO Model. Math. Anal. Numer., 25 (1991), 643-670. |
[6] |
G. D. Akrivis, V. D. Dougalis and V. A. Karakashian, On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation, Numer. Math., 59 (1991), 31-53.
doi: 10.1007/BF01385769. |
[7] |
Sh. Amiranashvili, U. Bandelow and A. Mielke, Padé approximant for refractive index and nonlocal envelope equations, Opt. Comm., 283 (2010), 480-485.
doi: 10.1016/j.optcom.2009.10.034. |
[8] |
Sh. Amiranashvili and A. Demircan, Hamiltonian structure of propagation equations for ultrashort optical pulses, Phys. Rev. A, 82 (2010), 013812.
doi: 10.1103/PhysRevA.82.013812. |
[9] |
Sh. Amiranashvili and A. Demircan, Ultrashort optical pulse propagation in terms of analytic signal, Advances in Optical Technologies, 2011 (2011), 989515, 8pp.
doi: 10.1155/2011/989515. |
[10] |
X. Antoine, A. Arnold, Ch. Besse, M. Ehrhardt and A. Schädle, A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations, Commun. Comput. Physics, 4 (2008), 729-796. |
[11] |
X. Antoine, W. Bao and C. Besse, Computational methods for the dynamics of nonlinear schrödinger/gross-pitaevskii equations, Comput. Phys. Commun., 184 (2013), 2621-2633.
doi: 10.1016/j.cpc.2013.07.012. |
[12] |
W. Bao, Sh. Jin and P. A. Markowich, Numerical study of time-splitting spectral discretizations of nonlinear Schrödinger equations in the semiclassical regimes, SIAM J. Sci. Comput., 25 (2003), 27-64.
doi: 10.1137/S1064827501393253. |
[13] |
W. Bao, Q. Tang and Z. Xu, Numerical methods and comparison for computing dark and bright solitons in the nonlinear schrödinger equation, J. Comput. Phys., 235 (2013), 423-445.
doi: 10.1016/j.jcp.2012.10.054. |
[14] |
M. Bass, E. W. Van Stryland, D. R. Williams and W. L. Wolfe, editors, Handbook of Optics, Volume 1, $2^{nd}$ edition, McGraw-Hill, 1995. |
[15] |
A. Borzi and E. Decker, Analysis of a leap-frog pseudospectral scheme for the Schrödinger equation, J. Comput. Appl. Math., 193 (2006), 65-88.
doi: 10.1016/j.cam.2005.04.066. |
[16] |
R. W. Boyd, Nonlinear Optics, $3^rd$ edition, Academic, New York, 2008. |
[17] |
Q. Chang, E. Jia and W. Sun, Difference schemes for solving the generalized nonlinear schrödinger equation, J. Comput. Phys., 148 (1999), 397-415.
doi: 10.1006/jcph.1998.6120. |
[18] |
R. Čiegis, I. Laukaitytė and M. Radziunas, Numerical algorithms for Schrödinger equations with artificial boundary conditions, Numer. Funct. Anal. Optim., 30 (2009), 903-923.
doi: 10.1080/01630560903393097. |
[19] |
R. Čiegis and M. Radziunas, Effective numerical integration of traveling wave model for edge-emitting broad-area semiconductor lasers and amplifiers, Math. Model. Anal., 15 (2010), 409-430.
doi: 10.3846/1392-6292.2010.15.409-430. |
[20] |
A. Demircan, Sh. Amiranashvili, C. Brée and G. Steinmeyer, Compressible octave spanning supercontinuum generation by two-pulse collisions, Phys. Rev. Lett., 110 (2013), 233901.
doi: 10.1103/PhysRevLett.110.233901. |
[21] |
A. Demircan, Sh. Amiranashvili and G. Steinmeyer, Controlling light by light with an optical event horizon, Phys. Rev. Lett., 106 (2011), 163901.
doi: 10.1103/PhysRevLett.106.163901. |
[22] |
J. M. Dudley, G. Genty and S. Coen, Supercontinuum generation in photonic crystal fiber, Rev. Mod. Phys., 78 (2006), 1135-1184.
doi: 10.1103/RevModPhys.78.1135. |
[23] |
M. D. Feit, J. A. Fleck and A. Steiger, Solution of the Schrödinger equation by a spectral method, J. Comput. Phys., 47 (1982), 412-433.
doi: 10.1016/0021-9991(82)90091-2. |
[24] | |
[25] |
W. Hundsdorfer and J. G. Verwer, Numerical Solution of Time-Dependent Advection-Difusion-Reaction Equations, Springer Series in Computational Mathematics, 33, Springer, Berlin, Heidelberg, New York, Tokyo, 2003.
doi: 10.1007/978-3-662-09017-6. |
[26] |
A. J. Lichtenberg and M. A. Lieberman, Regular and Chaotic Dynamics, Applied Mathematical Sciences, 38, $2^{nd}$ edition, Springer, 1992.
doi: 10.1007/978-1-4757-2184-3. |
[27] |
C. Lubich, On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations, Math. Comput., 77 (2008), 2141-2153.
doi: 10.1090/S0025-5718-08-02101-7. |
[28] |
G. Muslu and H. Erbay, Higher-order split-step Fourier schemes for the generalized nonlinear Schrödinger equation, Mathematics and Computers in Simulation, 67 (2005), 581-595.
doi: 10.1016/j.matcom.2004.08.002. |
[29] |
Ch. Neuhauser and M. Thalhammer, On the convergence of splitting methods for linear evolutionary Schrödinger equations involving an unbounded potential, BIT Numer. Math., 49 (2009), 199-215.
doi: 10.1007/s10543-009-0215-2. |
[30] |
D. Pathria and J. L. Morris, Exact solutions for a generalized nonlinear Schrödinger equation, Phys. Scripta, 39 (1989), 673-680.
doi: 10.1088/0031-8949/39/6/001. |
[31] |
D. Pathria and J. L. Morris, Pseudo-spectral solution of nonlinear Schrödinger equations, J. Comput. Phys., 87 (1990), 108-125.
doi: 10.1016/0021-9991(90)90228-S. |
[32] |
M. Radziunas, R. Čiegis and A. Mirinavičius, On compact high order finite difference schemes for linear Schrödinger problem on non-uniform meshes, Int. Journal of Num. Anal. Model., 11 (2014), 303-314. |
[33] |
J. M. Sanz-Serna and J. G. Verwer, Conservative and non-conservative schemes for the solution of the nonlinear Schrödinger equation, IMA J. Numer. Anal., 6 (1986), 25-42.
doi: 10.1093/imanum/6.1.25. |
[34] |
J. M. Sanz-Serna and M. P. Calvo, Numerical Hamiltonian Problems, Applied Mathematics and Mathematical Computation, 7, Chapman & Hall, London, 1994. |
[35] |
J. M. Stone and J. C. Knight, Visibly white light generation in uniform photonic crystal fiber using a microchip laser, Opt. Express, 16 (2008), 2670-2675.
doi: 10.1364/OE.16.002670. |
[36] |
G. Strang, On the construction and comparison of difference schemes, SIAM J. Numer. Anal., 5 (1968), 506-517.
doi: 10.1137/0705041. |
[37] |
T. R. Taha and M. J. Ablovitz, Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical Schrödinger equation, J. Comp. Phys., 55 (1984), 203-230.
doi: 10.1016/0021-9991(84)90003-2. |
[38] |
M. Thalhammer, M. Caliari and Ch. Neuhauser, High-order time-splitting Hermite and Fourier spectral methods, J. Comput. Phys., 228 (2009), 822-832.
doi: 10.1016/j.jcp.2008.10.008. |
[39] |
D. E. Vakman and L. A. Vainshtein, Amplitude, phase, frequency - fundamental concepts of oscillation theory, Usp. Fiz. Nauk, 20 (1977), 1002-1016.
doi: 10.1070/PU1977v020n12ABEH005479. |
[40] |
P. K. A. Wai, H. H. Chen and Y. C. Lee, Radiations by "solitons" at the zero group-dispersion wavelength of single-mode optical fibers, Phys. Rev. A, 41 (1990), 426-439.
doi: 10.1103/PhysRevA.41.426. |
[41] |
V. E. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Sov. Phys. JETP, 34 (1972), 62-69. |
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