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.