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

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

Received  August 2013 Revised  September 2014 Published  March 2015

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.
Citation: Shalva Amiranashvili, Raimondas  Čiegis, Mindaugas Radziunas. Numerical methods for a class of generalized nonlinear Schrödinger equations. Kinetic & 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.  doi: 10.1140/epjst/e2009-01072-0.  Google Scholar

[2]

G. P. Agrawal, Nonlinear Fiber Optics,, $4^{th}$ edition, (2007).   Google Scholar

[3]

N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams,, Chapman and Hall, (1997).   Google Scholar

[4]

N. Akhmediev and M. Karlsson, Cherenkov radiation emitted by solitons in optical fibers,, Phys. Rev. A, 51 (1995), 2602.  doi: 10.1103/PhysRevA.51.2602.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1007/BF01385769.  Google Scholar

[7]

Sh. Amiranashvili, U. Bandelow and A. Mielke, Padé approximant for refractive index and nonlocal envelope equations,, Opt. Comm., 283 (2010), 480.  doi: 10.1016/j.optcom.2009.10.034.  Google Scholar

[8]

Sh. Amiranashvili and A. Demircan, Hamiltonian structure of propagation equations for ultrashort optical pulses,, Phys. Rev. A, 82 (2010).  doi: 10.1103/PhysRevA.82.013812.  Google Scholar

[9]

Sh. Amiranashvili and A. Demircan, Ultrashort optical pulse propagation in terms of analytic signal,, Advances in Optical Technologies, 2011 (2011).  doi: 10.1155/2011/989515.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/j.cpc.2013.07.012.  Google Scholar

[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.  doi: 10.1137/S1064827501393253.  Google Scholar

[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.  doi: 10.1016/j.jcp.2012.10.054.  Google Scholar

[14]

M. Bass, E. W. Van Stryland, D. R. Williams and W. L. Wolfe, editors, Handbook of Optics, Volume 1,, $2^{nd}$ edition, (1995).   Google Scholar

[15]

A. Borzi and E. Decker, Analysis of a leap-frog pseudospectral scheme for the Schrödinger equation,, J. Comput. Appl. Math., 193 (2006), 65.  doi: 10.1016/j.cam.2005.04.066.  Google Scholar

[16]

R. W. Boyd, Nonlinear Optics,, $3^rd$ edition, (2008).   Google Scholar

[17]

Q. Chang, E. Jia and W. Sun, Difference schemes for solving the generalized nonlinear schrödinger equation,, J. Comput. Phys., 148 (1999), 397.  doi: 10.1006/jcph.1998.6120.  Google Scholar

[18]

R. Čiegis, I. Laukaitytė and M. Radziunas, Numerical algorithms for Schrödinger equations with artificial boundary conditions,, Numer. Funct. Anal. Optim., 30 (2009), 903.  doi: 10.1080/01630560903393097.  Google Scholar

[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.  doi: 10.3846/1392-6292.2010.15.409-430.  Google Scholar

[20]

A. Demircan, Sh. Amiranashvili, C. Brée and G. Steinmeyer, Compressible octave spanning supercontinuum generation by two-pulse collisions,, Phys. Rev. Lett., 110 (2013).  doi: 10.1103/PhysRevLett.110.233901.  Google Scholar

[21]

A. Demircan, Sh. Amiranashvili and G. Steinmeyer, Controlling light by light with an optical event horizon,, Phys. Rev. Lett., 106 (2011).  doi: 10.1103/PhysRevLett.106.163901.  Google Scholar

[22]

J. M. Dudley, G. Genty and S. Coen, Supercontinuum generation in photonic crystal fiber,, Rev. Mod. Phys., 78 (2006), 1135.  doi: 10.1103/RevModPhys.78.1135.  Google Scholar

[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.  doi: 10.1016/0021-9991(82)90091-2.  Google Scholar

[24]

A. Hasegawa, Optical Solitons in Fibers,, Springer, (1980).   Google Scholar

[25]

W. Hundsdorfer and J. G. Verwer, Numerical Solution of Time-Dependent Advection-Difusion-Reaction Equations,, Springer Series in Computational Mathematics, (2003).  doi: 10.1007/978-3-662-09017-6.  Google Scholar

[26]

A. J. Lichtenberg and M. A. Lieberman, Regular and Chaotic Dynamics,, Applied Mathematical Sciences, (1992).  doi: 10.1007/978-1-4757-2184-3.  Google Scholar

[27]

C. Lubich, On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations,, Math. Comput., 77 (2008), 2141.  doi: 10.1090/S0025-5718-08-02101-7.  Google Scholar

[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.  doi: 10.1016/j.matcom.2004.08.002.  Google Scholar

[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.  doi: 10.1007/s10543-009-0215-2.  Google Scholar

[30]

D. Pathria and J. L. Morris, Exact solutions for a generalized nonlinear Schrödinger equation,, Phys. Scripta, 39 (1989), 673.  doi: 10.1088/0031-8949/39/6/001.  Google Scholar

[31]

D. Pathria and J. L. Morris, Pseudo-spectral solution of nonlinear Schrödinger equations,, J. Comput. Phys., 87 (1990), 108.  doi: 10.1016/0021-9991(90)90228-S.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1093/imanum/6.1.25.  Google Scholar

[34]

J. M. Sanz-Serna and M. P. Calvo, Numerical Hamiltonian Problems,, Applied Mathematics and Mathematical Computation, (1994).   Google Scholar

[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.  doi: 10.1364/OE.16.002670.  Google Scholar

[36]

G. Strang, On the construction and comparison of difference schemes,, SIAM J. Numer. Anal., 5 (1968), 506.  doi: 10.1137/0705041.  Google Scholar

[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.  doi: 10.1016/0021-9991(84)90003-2.  Google Scholar

[38]

M. Thalhammer, M. Caliari and Ch. Neuhauser, High-order time-splitting Hermite and Fourier spectral methods,, J. Comput. Phys., 228 (2009), 822.  doi: 10.1016/j.jcp.2008.10.008.  Google Scholar

[39]

D. E. Vakman and L. A. Vainshtein, Amplitude, phase, frequency - fundamental concepts of oscillation theory,, Usp. Fiz. Nauk, 20 (1977), 1002.  doi: 10.1070/PU1977v020n12ABEH005479.  Google Scholar

[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.  doi: 10.1103/PhysRevA.41.426.  Google Scholar

[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.   Google Scholar

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.  doi: 10.1140/epjst/e2009-01072-0.  Google Scholar

[2]

G. P. Agrawal, Nonlinear Fiber Optics,, $4^{th}$ edition, (2007).   Google Scholar

[3]

N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams,, Chapman and Hall, (1997).   Google Scholar

[4]

N. Akhmediev and M. Karlsson, Cherenkov radiation emitted by solitons in optical fibers,, Phys. Rev. A, 51 (1995), 2602.  doi: 10.1103/PhysRevA.51.2602.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1007/BF01385769.  Google Scholar

[7]

Sh. Amiranashvili, U. Bandelow and A. Mielke, Padé approximant for refractive index and nonlocal envelope equations,, Opt. Comm., 283 (2010), 480.  doi: 10.1016/j.optcom.2009.10.034.  Google Scholar

[8]

Sh. Amiranashvili and A. Demircan, Hamiltonian structure of propagation equations for ultrashort optical pulses,, Phys. Rev. A, 82 (2010).  doi: 10.1103/PhysRevA.82.013812.  Google Scholar

[9]

Sh. Amiranashvili and A. Demircan, Ultrashort optical pulse propagation in terms of analytic signal,, Advances in Optical Technologies, 2011 (2011).  doi: 10.1155/2011/989515.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/j.cpc.2013.07.012.  Google Scholar

[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.  doi: 10.1137/S1064827501393253.  Google Scholar

[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.  doi: 10.1016/j.jcp.2012.10.054.  Google Scholar

[14]

M. Bass, E. W. Van Stryland, D. R. Williams and W. L. Wolfe, editors, Handbook of Optics, Volume 1,, $2^{nd}$ edition, (1995).   Google Scholar

[15]

A. Borzi and E. Decker, Analysis of a leap-frog pseudospectral scheme for the Schrödinger equation,, J. Comput. Appl. Math., 193 (2006), 65.  doi: 10.1016/j.cam.2005.04.066.  Google Scholar

[16]

R. W. Boyd, Nonlinear Optics,, $3^rd$ edition, (2008).   Google Scholar

[17]

Q. Chang, E. Jia and W. Sun, Difference schemes for solving the generalized nonlinear schrödinger equation,, J. Comput. Phys., 148 (1999), 397.  doi: 10.1006/jcph.1998.6120.  Google Scholar

[18]

R. Čiegis, I. Laukaitytė and M. Radziunas, Numerical algorithms for Schrödinger equations with artificial boundary conditions,, Numer. Funct. Anal. Optim., 30 (2009), 903.  doi: 10.1080/01630560903393097.  Google Scholar

[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.  doi: 10.3846/1392-6292.2010.15.409-430.  Google Scholar

[20]

A. Demircan, Sh. Amiranashvili, C. Brée and G. Steinmeyer, Compressible octave spanning supercontinuum generation by two-pulse collisions,, Phys. Rev. Lett., 110 (2013).  doi: 10.1103/PhysRevLett.110.233901.  Google Scholar

[21]

A. Demircan, Sh. Amiranashvili and G. Steinmeyer, Controlling light by light with an optical event horizon,, Phys. Rev. Lett., 106 (2011).  doi: 10.1103/PhysRevLett.106.163901.  Google Scholar

[22]

J. M. Dudley, G. Genty and S. Coen, Supercontinuum generation in photonic crystal fiber,, Rev. Mod. Phys., 78 (2006), 1135.  doi: 10.1103/RevModPhys.78.1135.  Google Scholar

[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.  doi: 10.1016/0021-9991(82)90091-2.  Google Scholar

[24]

A. Hasegawa, Optical Solitons in Fibers,, Springer, (1980).   Google Scholar

[25]

W. Hundsdorfer and J. G. Verwer, Numerical Solution of Time-Dependent Advection-Difusion-Reaction Equations,, Springer Series in Computational Mathematics, (2003).  doi: 10.1007/978-3-662-09017-6.  Google Scholar

[26]

A. J. Lichtenberg and M. A. Lieberman, Regular and Chaotic Dynamics,, Applied Mathematical Sciences, (1992).  doi: 10.1007/978-1-4757-2184-3.  Google Scholar

[27]

C. Lubich, On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations,, Math. Comput., 77 (2008), 2141.  doi: 10.1090/S0025-5718-08-02101-7.  Google Scholar

[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.  doi: 10.1016/j.matcom.2004.08.002.  Google Scholar

[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.  doi: 10.1007/s10543-009-0215-2.  Google Scholar

[30]

D. Pathria and J. L. Morris, Exact solutions for a generalized nonlinear Schrödinger equation,, Phys. Scripta, 39 (1989), 673.  doi: 10.1088/0031-8949/39/6/001.  Google Scholar

[31]

D. Pathria and J. L. Morris, Pseudo-spectral solution of nonlinear Schrödinger equations,, J. Comput. Phys., 87 (1990), 108.  doi: 10.1016/0021-9991(90)90228-S.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1093/imanum/6.1.25.  Google Scholar

[34]

J. M. Sanz-Serna and M. P. Calvo, Numerical Hamiltonian Problems,, Applied Mathematics and Mathematical Computation, (1994).   Google Scholar

[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.  doi: 10.1364/OE.16.002670.  Google Scholar

[36]

G. Strang, On the construction and comparison of difference schemes,, SIAM J. Numer. Anal., 5 (1968), 506.  doi: 10.1137/0705041.  Google Scholar

[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.  doi: 10.1016/0021-9991(84)90003-2.  Google Scholar

[38]

M. Thalhammer, M. Caliari and Ch. Neuhauser, High-order time-splitting Hermite and Fourier spectral methods,, J. Comput. Phys., 228 (2009), 822.  doi: 10.1016/j.jcp.2008.10.008.  Google Scholar

[39]

D. E. Vakman and L. A. Vainshtein, Amplitude, phase, frequency - fundamental concepts of oscillation theory,, Usp. Fiz. Nauk, 20 (1977), 1002.  doi: 10.1070/PU1977v020n12ABEH005479.  Google Scholar

[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.  doi: 10.1103/PhysRevA.41.426.  Google Scholar

[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.   Google Scholar

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