Numerical methods for a class of generalized nonlinear Schrödinger equations

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June 2015, 8(2): 215-234. doi: 10.3934/krm.2015.8.215

Numerical methods for a class of generalized nonlinear Schrödinger equations

Shalva Amiranashvili ^1, , Raimondas Čiegis ^2, and Mindaugas Radziunas ^1,

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

Received August 2013 Revised September 2014 Published March 2015

Abstract
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We present and analyze different splitting algorithms for numerical solution of the both classical and generalized nonlinear Schrödinger equations describing propagation of wave packets with special emphasis on applications to nonlinear fiber-optics. The considered generalizations take into account the higher-order corrections of the linear differential dispersion operator as well as the saturation of nonlinearity and the self-steepening of the field envelope function. For stabilization of the pseudo-spectral splitting schemes for generalized Schrödinger equations a regularization based on the approximation of the derivatives by the low number of Fourier modes is proposed. To illustrate the theoretically predicted performance of these schemes several numerical experiments have been done. In particular, we compute real-world examples of extreme pulses propagating in silica fibers.

Keywords: finite diiference method, spectral method., splitting method, Generalized nonlinear Schrödinger equations.

Mathematics Subject Classification: Primary: 65M60, 65M70; Secondary: 35G3.

Citation: Shalva Amiranashvili, Raimondas Čiegis, Mindaugas Radziunas. Numerical methods for a class of generalized nonlinear Schrödinger equations. Kinetic and Related Models, 2015, 8 (2) : 215-234. doi: 10.3934/krm.2015.8.215

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]	A. Hasegawa, Optical Solitons in Fibers, Springer, 1980.
[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]	A. Hasegawa, Optical Solitons in Fibers, Springer, 1980.
[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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