Research Article Open Access

Dynamics of Coherent Structures in the Coupled Complex Ginzburg-Landau Equations

Tat-Leung Yee1
  • 1 , Hong Kong
Journal of Mathematics and Statistics
Volume 8 No. 3, 2012, 413-418

DOI: https://doi.org/10.3844/jmssp.2012.413.418

Submitted On: 11 July 2012 Published On: 6 November 2012

How to Cite: Yee, T. (2012). Dynamics of Coherent Structures in the Coupled Complex Ginzburg-Landau Equations. Journal of Mathematics and Statistics, 8(3), 413-418. https://doi.org/10.3844/jmssp.2012.413.418

Abstract

Problem statement: In this study, we study the analytical construction of some exact solutions of a system of coupled physical differential equations, namely, the Complex Ginzburg-Landau Equations (CGLEs). CGLEs are intensively studied models of pattern formation in nonlinear dissipative media, with applications to biology, hydrodynamics, nonlinear optics, plasma physics, reaction-diffusion systems and many other fields. Approach: A system of two coupled CGLEs modeling the propagation of pulses under the combined influence of dispersion, self and cross phase modulations, linear and nonlinear gain and loss will be discussed. A Solitary Pulse (SP) is a localized wave form and a front (also termed as shock) refers to a transition connecting two constant, but unequal, asymptotic states. A SP-front pair solution can be analytically obtained by the modified Hirota bilinear method. Results: These wave solutions are deduced by a system of six nonlinear algebraic equations, allowing the amplitudes, wave-numbers, frequency and velocities to be determined. Conclusion: The final exact solution can then be computed by applying the Groebner basis method with a large amount of algebraic simplifications done by the computer software Maple.

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Keywords

  • Solitary pulses and fronts
  • modified hirota bilinear method
  • complex ginzburg-landau equations