The discrete complex Ginzburg-Landau equation is a fundamental model for the dynamics of nonlinear lattices incorporating competitive dissipation and energy gain effects. Such mechanisms are particular importance study survival/destruction localised structures in many physical situations. In this work, we prove that dissipative solitonic waveforms persist significant times by introducing dynami...

Equations built on fractional derivatives prove to be a powerful tool in the description of complex systems when the effects of singularity, fractal supports, and long-range dependence play a role. In this paper, we advocate an application of the fractional derivative formalism to a fairly general class of critical phenomena when the organization of the system near the phase transition point is...

We present some mathematical aspects of Landau-Ginzburg string vacua in terms of toric geometry. The one-to-one correspondence between toric divisors and some of (−1, 1) states in Landau-Ginzburg model is presented for superpotentials of typical types. The Landau-Ginzburg interpretation of non-toric divisors is also presented. Using this interpretation, we propose a method to solve the so-calle...

Abstract Oscillons are spatially localized, time-periodic structures that have been observed in many natural processes, often under temporally periodic forcing. Near Hopf bifurcations, such systems can be formally reduced to forced complex Ginzburg–Landau equations, with oscillons then corresponding to stationary localized patterns. In this manuscript, stationary localized structures of the pla...

We show numerically that the one-dimensional quintic complex Ginzburg–Landau equation admits four different types of stable hole solutions. We present a simple analytic method which permits to calculate the region of existence and approximate shape of stable hole solutions in this equation. The analytic results are in good agreement with numerical simulations.

We study a complex Ginzburg-Landau equation in the plane, which has the form of a Gross-Pitaevskii equation with some dissipation added. We focus on the regime corresponding to well-prepared unitary vortices and derive their asymptotic motion law. 2000 Mathematics Subject Classification: 35B20,35B40,35Q40,82D55.

We reformulate the one-dimensional complex Ginzburg-Landau equation as a fourth order ordinary differential equation in order to find stationary spatiallyperiodic solutions. Using this formalism, we prove the existence and stability of stationary modulated-amplitude wave solutions. Approximate analytic expressions and a comparison with numerics are given.

We have found new dissipative solitons of the complex cubic-quintic Ginzburg-Landau equation with extreme amplitudes and short duration. At certain range of the equation parameters, these extreme spikes appear in pairs of slightly unequal amplitude. The bifurcation diagram of pulse amplitude versus dispersion parameter is constructed. c © 2015 Optical Society of America OCIS codes: 060.5530, 14...