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An Efficient Numerical Method for the Quintic Complex Swift-Hohenberg Equation

Published online by Cambridge University Press:  28 May 2015

Hanquan Wang  and
Lina Yanti
Hanquan Wang*
Affiliation:
School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan Province, 650221, P. R. China. Department of Mathematics, National University of Singapore, 117543, Singapore
Lina Yanti*
Affiliation:
Department of Mathematics, National University of Singapore, 117543, Singapore
*
Corresponding author.Email address:hanquan.wang@gmail.com
Corresponding author.Email address:linayanti@nus.edu.sg
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Abstract

In this paper, we present an efficient time-splitting Fourier spectral method for the quintic complex Swift-Hohenberg equation. Using the Strang time-splitting technique, we split the equation into linear part and nonlinear part. The linear part is solved with Fourier Pseudospectral method; the nonlinear part is solved analytically. We show that the method is easy to be applied and second-order in time and spectrally accurate in space. We apply the method to investigate soliton propagation, soliton interaction, and generation of stable moving pulses in one dimension and stable vortex solitons in two dimensions.

Keywords

65M7065Z05Quintic complex Swift-Hohenberg equationtime-splitting Fourier pseudospectral methodnumerical simulationsoliton
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 4 , Issue 2 , May 2011 , pp. 237 - 254
Copyright
Copyright © Global Science Press Limited 2011

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References

[1]Agrawal, G. P., Optical pulse propagation in doped fiber amplifiers, Phys. Rev. A, 44 (1991), pp. 7493.CrossRefGoogle ScholarPubMed
[2]Akhmediev, N., Rodrigues, A. S., and Town, G. E., Interaction of dual-frequency pulsed in passively mode-locked lasers, Opt. Comm., 187(2001), pp. 419–426.CrossRefGoogle Scholar
[3]Akhmediev, N., Soto-Crespo, J. M., and Town, G., Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: complex Ginzburg-Landau equation approach, Phys. Rev. E, 63 (2001), pp. 056602.CrossRefGoogle ScholarPubMed
[4]Aranson, I., Hochheiser, D., and Moloney, J. V., Boundary-driven selection of patterns in large-aspect-ratio lasers, Phys. Rev. A, 55(1997), pp. 3173.CrossRefGoogle Scholar
[5]Bao, W. and Jaksch, D., An explicit unconditionally stable numerical method for solving damped nonlinear Schrodinger equations with a focusing nonlinearity, SIAM J. Numer. Anal, 41(2003), pp. 1406–1426.CrossRefGoogle Scholar
[6]Bao, W., Jin, S., and Markowich, P. A., On time-splitting spectral approximations for the Schrodinger equation in the semiclasslcal regime, J. Comput. Phys., 175 (2002), pp. 487–524.CrossRefGoogle Scholar
[7]Bao, W. and Zhang, Y., Dynamics of the ground state and central vortex states in Bose-Einstein condensation, Math. Mod. Meth. Appl. Sci., 15(2005), pp. 1863–1896.CrossRefGoogle Scholar
[8]Burke, J. and Knobloch, E., Localized states in the generalied Swift-Hollenberg equation, Phy. Rev. E. 73(2006), pp. 056211.CrossRefGoogle Scholar
[9]Caginalp, G. and Fife, P. C., Higher order phase field models and detailed anisotropv, Phys. Rev. B, 34(1986), pp. 4940–4943.CrossRefGoogle Scholar
[10]Crawford, C. and Riecke, H., Oscillon-type structures and their interactlon in a Swift-Hohenberg model, Physica D, 129(1999), pp. 83–92.CrossRefGoogle Scholar
[11]Christov, C. I. and Pontes, J., Numerical scheme for Swift-Hohenberg equation with strict implementation of Lyapunov Functional, Mathematical and Computer Modelling 35 (2002) 87–99.CrossRefGoogle Scholar
[12]Crasovan, L.-C., Malomed, B. A., and Mihalache, D., Stable vortex solitons in the two-dimensional Ginzburg-Landau equation, Phys. Rev. E., 63, pp. 016605.Google Scholar
[13]Dee, G. T. and Van Saarloos, W., Bistable systems with propagating fronts leading to pattern formation, Phys. Rev. Lett., 60(1988), pp. 2641–2644.CrossRefGoogle ScholarPubMed
[14]Gardner, R. A. and Jones, C. K. R. T., Travelling waves of a perturbed diffusion equation arising in a phase field model, Indiana Univ. Math. J, 38(1989), pp. 1197–1222.Google Scholar
[15]Hat, Q. and Mazenb, G. F., Growth of order in an anlsotronlc Swift-Hohenberg model, Phys. Rev. E 73(2006), pp. 036117.Google Scholar
[16]Herrero, H. and Riecke, H., Bound pairs of fronts in a real Ginzburg-Landau equation coupled to a mean field, Physica D, 85(1995), pp. 79–92.CrossRefGoogle Scholar
[17]Kolodner, P., Drift, shape, and intrinsic destabillzation of pulses of traveling-wave convection, Phys. Rev, A, 44(1991), pp. 6448.CrossRefGoogle ScholarPubMed
[18]Kuramoto, Y., Chemical oscillations, Waves and turbulence, Springer-Verlag, Berlin. 1984.CrossRefGoogle Scholar
[19]Lega, J., Moloney, J. V., and Newell, A. C., Swift-Hohenberg equation for lasers, Phys. Rev Lett., 73(1994), pp. 2978.CrossRefGoogle ScholarPubMed
[20]Longhi, S. and Geraci, A., Swift-Hohenberz equation for optical parametrlc oscillators, Phys. Rev. A, 54(1996), pp. 4581.CrossRefGoogle ScholarPubMed
[21]Maruno, K. I., Ankiewicz, A., and Akhmediev, N., Exact soliton solutions of the one-dimensional complex Swift-Hohenberg equation, Physica D, 176(2003), pp. 44–66.CrossRefGoogle Scholar
[22]Mercier, J. F. and Moloney, J. V., Derivation of semiconductor laser mean-field and Swift-Hohenberg equations,Phys. Rev. E, 66(2002), pp. 036221.CrossRefGoogle ScholarPubMed
[23]Peletier, L. A. and Rottschafer, V., Pattern selection of the solutions of the Swift-Hohenberg equation, Physica D, 194(2004), pp. 95–126.CrossRefGoogle Scholar
[24]Qian, H. and Mazenko, G. F., Growth of order in an anisotropic Swift-Hohenberg model, Phys. Rev. E, 73(2006), pp. 036117.CrossRefGoogle Scholar
[25]Sakaguchi, H., Herringbone Pattern for an anisotropic complex Swift-Hohenberg equation, Phys. Rev. E, 58(1998), pp. 8021–8023.CrossRefGoogle Scholar
[26]Sakaguchi, H., Motion of pulses and vortices in the cublc-quintic complex Ginzburg-Landau equation without viscosity, Physica D, 210(2005), pp. 138–148.CrossRefGoogle Scholar
[27]Sakaguchi, H. and Brand, H. R., Localized patterns for the quintic complex Swift-Hohenberg equation, Physica D, 117(1998), pp. 95–105.CrossRefGoogle Scholar
[28]Sakaguchi, H. and Brand, H. R., Stable localized solutions of arbitrary length for the quintic Swift-Hohenberg equation, Physica D, 97(1996), pp. 274–285.CrossRefGoogle Scholar
[29]Sanchez-Morcillo, V. J., Rolddan, E., G. J. de Valcarcel, and Staliunas, K., Generalized complex Swift-Hohenberg equation for optical parametrlc oscillators, Phys. Rev., A, 56(1997), pp. 3237.CrossRefGoogle Scholar
[30]Soto-Crespo, J. M. and Akhmediev, N., Composite solitons and two-pulse generation in passively mode-locked lasers modeled by the complex quintic Swift-Hohenberg equation, Phys. Rev. E. 66(2002), pp. 066610.CrossRefGoogle ScholarPubMed
[31]Soto-Crespo, J. M. and Akhmediev, N., and Afansjev, V. V., Stability of the pulselike solutions of the quintic complex Ginzburg-Landau equation, J. Opt. Soc. Am. B, 13(1996), pp. 1439–1449.CrossRefGoogle Scholar
[32]Yanti, L., Numerical studies of quintic complex Swift-Hohenberg equation, Thesis for the degree of Bachelor of Science with Honours, Department of Mathematics, National University of Singapore, 2007.Google Scholar