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The fractional complex cubic-quintic Ginzburg-Landau equation (FCCQGLE) with variable coefficients is a propagation model of optical pulses in optical fibers, the quintic term contained in it not only explains the physical significances that are not found in the existing models, but also has more abundant dynamic characteristics compared with lower dimensional systems. In this paper, the exact solutions of this equation are obtained via the appropriate transformations methods, which also are divided into different kinds. More precisely, we acquire the solitary, soliton and elliptic waves solutions by using the improved unified method, the bright and dark solitons solutions by using the improved F-expansion method, as well as the traveling wave solutions through the improved ( G ′ / G 2 ) -expansion and Bernoulli sub-equation function methods. Furthermore, we draw the 3D, 2D, density and contour plots to understand the propagation forms and the changes of amplitudes, frequencies and shapes of the solutions for the FCCQGLE with variable coefficients. The last thing worth mentioning is that the improved unified method here extends the degree of the polynomial from n to -n in the hypothetical solutions, which is not appeared before.
The fractional complex cubic-quintic Ginzburg-Landau equation (FCCQGLE) with variable coefficients is a propagation model of optical pulses in optical fibers, the quintic term contained in it not only explains the physical significances that are not found in the existing models, but also has more abundant dynamic characteristics compared with lower dimensional systems. In this paper, the exact solutions of this equation are obtained via the appropriate transformations methods, which also are divided into different kinds. More precisely, we acquire the solitary, soliton and elliptic waves solutions by using the improved unified method, the bright and dark solitons solutions by using the improved F-expansion method, as well as the traveling wave solutions through the improved ( G ′ / G 2 ) -expansion and Bernoulli sub-equation function methods. Furthermore, we draw the 3D, 2D, density and contour plots to understand the propagation forms and the changes of amplitudes, frequencies and shapes of the solutions for the FCCQGLE with variable coefficients. The last thing worth mentioning is that the improved unified method here extends the degree of the polynomial from n to -n in the hypothetical solutions, which is not appeared before.
When extending the complex Ginzburg–Landau equation (CGLE) to more than one spatial dimension, there is an underlying question of whether one is capturing all the interesting physics inherent in these higher dimensions. Although spatial anisotropy is far less studied than its isotropic counterpart, anisotropy is fundamental in applications to superconductors, plasma physics, and geology, to name just a few examples. We first formulate the CGLE on anisotropic, time‐varying media, with this time variation permitting a degree of control of the anisotropy over time, focusing on how time‐varying anisotropy influences diffusion and dispersion within both bounded and unbounded space domains. From here, we construct a variety of exact dissipative nonlinear wave solutions, including analogs of wavetrains, solitons, breathers, and rogue waves, before outlining the construction of more general solutions via a dissipative, nonautonomous generalization of the variational method. We finally consider the problem of modulational instability within anisotropic, time‐varying media, obtaining generalizations to the Benjamin–Feir instability mechanism. We apply this framework to study the emergence and control of anisotropic spatiotemporal chaos in rectangular and curved domains. Our theoretical framework and specific solutions all point to time‐varying anisotropy being a potentially valuable feature for the manipulation and control of waves in anisotropic media.
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