Attactors of dissipative soliton equation
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Dissipative soliton
We study the behavior of the soliton which, while moving in non-dissipative medium encounters a barrier with dissipation. The modelling included the case of a finite dissipative layer as well as a wave passing from a dissipative layer into a non-dissipative one and vice versa. New effects are presented in the case of numerically finite barrier on the soliton path: first, if the form of dissipation distribution has a form of a frozen soliton, the wave that leaves the dissipative barrier becomes a bi-soliton and a reflection wave arises as a comparatively small and quasi-harmonic oscillation. Second, if the dissipation is negative (the wave, instead of loosing energy, is pumped with it) the passed wave is a soliton of a greater amplitude and velocity. Third, when the travelling wave solution of the KdV-Burgers (it is a shock wave in a dissipative region) enters a non-dissipative layer this shock transforms into a quasi-harmonic oscillation known for the KdV.
Dissipative soliton
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Dissipative soliton
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We study the behavior of the soliton which, while moving in non-dissipative medium encounters a barrier with dissipation. The modelling included the case of a finite dissipative layer as well as a wave passing from a dissipative layer into a non-dissipative one and vice versa. New effects are presented in the case of numerically finite barrier on the soliton path: first, if the form of dissipation distribution has a form of a frozen soliton, the wave that leaves the dissipative barrier becomes a bi-soliton and a reflection wave arises as a comparatively small and quasi-harmonic oscillation. Second, if the dissipation is negative (the wave, instead of loosing energy, is pumped with it) the passed wave is a soliton of a greater amplitude and velocity. Third, when the travelling wave solution of the KdV-Burgers (it is a shock wave in a dissipative region) enters a non-dissipative layer this shock transforms into a quasi-harmonic oscillation known for the KdV.
Dissipative soliton
Harmonic
Oscillation (cell signaling)
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Dissipative soliton resonance (DSR) is a phenomenon where the energy of a soliton in a dissipative system increases without limit at certain values of the system parameters. Using the method of collective variable approach, we have found an approximate relation between the parameters of the normalized complex cubic-quintic Ginzburg-Landau equation where the resonance manifests itself. Comparisons between the results obtained by collective variable approach, and those obtained by the method of moments show good qualitative agreement. This choice also helps to see the influence of the active terms on the resonance curve, so can be very useful in constructing passively mode-locked laser that generate solitons with the highest possible energies.
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We have experimentally investigated the propagation of a soliton in a nonlinear LC circuit with inhomogeneity and dissipation. In a homogeneous and dissipative circuit, the amplitude of a soliton decreases with the distance. The damping of a soliton is weakened if the inhomogeneity is introduced in a dissipative circuit, and the amplitude begins to increase in the circuit of strong inhomogeneity. The disintegration of soliton is not clearly observed in the present experiment. The experimental result is qualitatively explained by the Korteweg-de Vries equation with inhomogeneity and dissipation.
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In the third and final talk on dissipative structures in fiber applications, we discuss mathematical techniques that can be used to characterize modern laser systems that consist of several discrete elements. In particular, we use a nonlinear mapping technique to evaluate high power laser systems where significant changes in the pulse evolution per cavity round trip is observed. We demonstrate that dissipative soliton solutions might be effectively described using this Poincare mapping approach.
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