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
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show all references

References:
[1]

Eur. Phys. J. Special Topics, 173 (2009), 147-166. doi: 10.1140/epjst/e2009-01072-0.  Google Scholar

[2]

$4^{th}$ edition, Academic, New York, 2007. Google Scholar

[3]

Chapman and Hall, 1997. Google Scholar

[4]

Phys. Rev. A, 51 (1995), 2602-2607. doi: 10.1103/PhysRevA.51.2602.  Google Scholar

[5]

RAIRO Model. Math. Anal. Numer., 25 (1991), 643-670.  Google Scholar

[6]

Numer. Math., 59 (1991), 31-53. doi: 10.1007/BF01385769.  Google Scholar

[7]

Opt. Comm., 283 (2010), 480-485. doi: 10.1016/j.optcom.2009.10.034.  Google Scholar

[8]

Phys. Rev. A, 82 (2010), 013812. doi: 10.1103/PhysRevA.82.013812.  Google Scholar

[9]

Advances in Optical Technologies, 2011 (2011), 989515, 8pp. doi: 10.1155/2011/989515.  Google Scholar

[10]

Commun. Comput. Physics, 4 (2008), 729-796.  Google Scholar

[11]

Comput. Phys. Commun., 184 (2013), 2621-2633. doi: 10.1016/j.cpc.2013.07.012.  Google Scholar

[12]

SIAM J. Sci. Comput., 25 (2003), 27-64. doi: 10.1137/S1064827501393253.  Google Scholar

[13]

J. Comput. Phys., 235 (2013), 423-445. doi: 10.1016/j.jcp.2012.10.054.  Google Scholar

[14]

$2^{nd}$ edition, McGraw-Hill, 1995. Google Scholar

[15]

J. Comput. Appl. Math., 193 (2006), 65-88. doi: 10.1016/j.cam.2005.04.066.  Google Scholar

[16]

$3^rd$ edition, Academic, New York, 2008.  Google Scholar

[17]

J. Comput. Phys., 148 (1999), 397-415. doi: 10.1006/jcph.1998.6120.  Google Scholar

[18]

Numer. Funct. Anal. Optim., 30 (2009), 903-923. doi: 10.1080/01630560903393097.  Google Scholar

[19]

Math. Model. Anal., 15 (2010), 409-430. doi: 10.3846/1392-6292.2010.15.409-430.  Google Scholar

[20]

Phys. Rev. Lett., 110 (2013), 233901. doi: 10.1103/PhysRevLett.110.233901.  Google Scholar

[21]

Phys. Rev. Lett., 106 (2011), 163901. doi: 10.1103/PhysRevLett.106.163901.  Google Scholar

[22]

Rev. Mod. Phys., 78 (2006), 1135-1184. doi: 10.1103/RevModPhys.78.1135.  Google Scholar

[23]

J. Comput. Phys., 47 (1982), 412-433. doi: 10.1016/0021-9991(82)90091-2.  Google Scholar

[24]

Springer, 1980. Google Scholar

[25]

Springer Series in Computational Mathematics, 33, Springer, Berlin, Heidelberg, New York, Tokyo, 2003. doi: 10.1007/978-3-662-09017-6.  Google Scholar

[26]

Applied Mathematical Sciences, 38, $2^{nd}$ edition, Springer, 1992. doi: 10.1007/978-1-4757-2184-3.  Google Scholar

[27]

Math. Comput., 77 (2008), 2141-2153. doi: 10.1090/S0025-5718-08-02101-7.  Google Scholar

[28]

Mathematics and Computers in Simulation, 67 (2005), 581-595. doi: 10.1016/j.matcom.2004.08.002.  Google Scholar

[29]

BIT Numer. Math., 49 (2009), 199-215. doi: 10.1007/s10543-009-0215-2.  Google Scholar

[30]

Phys. Scripta, 39 (1989), 673-680. doi: 10.1088/0031-8949/39/6/001.  Google Scholar

[31]

J. Comput. Phys., 87 (1990), 108-125. doi: 10.1016/0021-9991(90)90228-S.  Google Scholar

[32]

Int. Journal of Num. Anal. Model., 11 (2014), 303-314.  Google Scholar

[33]

IMA J. Numer. Anal., 6 (1986), 25-42. doi: 10.1093/imanum/6.1.25.  Google Scholar

[34]

Applied Mathematics and Mathematical Computation, 7, Chapman & Hall, London, 1994.  Google Scholar

[35]

Opt. Express, 16 (2008), 2670-2675. doi: 10.1364/OE.16.002670.  Google Scholar

[36]

SIAM J. Numer. Anal., 5 (1968), 506-517. doi: 10.1137/0705041.  Google Scholar

[37]

J. Comp. Phys., 55 (1984), 203-230. doi: 10.1016/0021-9991(84)90003-2.  Google Scholar

[38]

J. Comput. Phys., 228 (2009), 822-832. doi: 10.1016/j.jcp.2008.10.008.  Google Scholar

[39]

Usp. Fiz. Nauk, 20 (1977), 1002-1016. doi: 10.1070/PU1977v020n12ABEH005479.  Google Scholar

[40]

Phys. Rev. A, 41 (1990), 426-439. doi: 10.1103/PhysRevA.41.426.  Google Scholar

[41]

Sov. Phys. JETP, 34 (1972), 62-69.  Google Scholar

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