TY  - JOUR
AU  - Yee, Tat-Leung 
PY  - 2012
TI  - Dynamics of Coherent Structures in the Coupled Complex Ginzburg-Landau Equations
JF  - Journal of Mathematics and Statistics
VL  - 8
IS  - 3
DO  - 10.3844/jmssp.2012.413.418
UR  - https://thescipub.com/abstract/jmssp.2012.413.418
AB  - 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.