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Two-dimensional structures in the quintic Ginzburg–Landau equation

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By using ZEUS Linux cluster at Embry-Riddle Aeronautical University, we perform extensive numerical simulations based on a two-dimensional Fourier spectral method (Fourier spatial discretization combined with an explicit scheme for time differencing) to find the range of existence of the spatiotemporal solitons of the two-dimensional complex Ginzburg–Landau equation with cubic and quintic nonlinearities. We start from the system parameters used previously by Akhmediev and slowly vary them one by one to determine the regimes where solitons exist as stable/unstable structures. We present eight classes of dissipative solitons from which six are known (stationary, pulsating, vortex spinning rings, filament, exploding, creeping) and two are novel (creeping-vortex “propellers” and spinning “bean-shaped” solitons). By running lengthy simulations for the different parameters of the equation, we find ranges of existence of stable structures (stationary, pulsating, circular vortex spinning, organized exploding) and unstable structures (circularvortex spinning that leads to filament, disorganized exploding, and creeping). Moreover, by varying the two initial conditions together with vorticity, we find a richer behavior in the form of creeping-vortex “propellers” and spinning “bean-shaped” solitons. Each class differentiates from the other by distinctive features of their energy evolution, shape of initial conditions, as well as domain of existence of parameters.

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Authors and Affiliations

  1. Department of Mathematics, ENSEIRB-MATMECA, Université Bordeaux 1, 33405, Bordeaux, France

    Florent Bérard & Charles-Julien Vandamme

  2. Department of Mathematics, Embry-Riddle Aeronautical University, Daytona Beach, FL, 32114-3900, USA

    Stefan C. Mancas

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  1. Florent Bérard
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  3. Stefan C. Mancas
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Correspondence to Stefan C. Mancas.

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Bérard, F., Vandamme, CJ. & Mancas, S.C. Two-dimensional structures in the quintic Ginzburg–Landau equation. Nonlinear Dyn 81, 1413–1433 (2015). https://doi.org/10.1007/s11071-015-2077-2

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