Dynamics of Coherent Structures in the Coupled Complex Ginzburg-Landau Equations
DOI : 10.3844/jmssp.2012.413.418
Journal of Mathematics and Statistics
Volume 8, Issue 3
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.
© 2012 Tat-Leung Yee. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.