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.
Highlights
The ‘bright soliton-front’ pair of Complex GinzburgLandau Equations (CGLEs) can be obtained by rewriting the partial differential equations as ‘trilinear’ forms with the Bekki-Nozaki modified Hirota operator
Varying amplitudes of the electric fields A and B will typically be governed by the nonlinearly coupled CGLEs Eq 6: case where gain/loss are absent, i.e., CGLEs reduce to the coupled nonlinear Schrödinger equations
A two-waveguide system including the gain/loss is governed by the coupled CGLEs and one model of nonlinear coupling is investigated in this study
Summary
The dynamics and propagation of the pulses are governed by the combined influence of dispersion, self and cross phase modulations, linear and nonlinear gain/loss. The primary focus in the study is a system of two waveguides governed by two coupled CGLEs. Conditions for the presence of a shock/wave front in one channel and a bright Solitary Pulse (SP) in the other, will be elucidated. This can be illustrated by a simple case where damping and gain are absent, i.e., CGLEs are reduced to the integrable, nonlinear Schrödinger (Manakov) equations.
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