Ionization fronts and their velocities in a coupled Ginzburg-Landau model

Bruhn, B.

doi:10.1063/1.2173953

Cited by 5 publications

(2 citation statements)

References 21 publications

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“…It is worthwhile to mention that similar equations have been obtained earlier by Janssen and Rasmussen 7 . Although in a different physical context, amplitude equations similar to the ones derived in the previous section have been reported by B. Bruhn et al 15,16 . Recently, numerical simulations of a discrete self-consistent waveparticle model describing an ensemble of particles coupled to a monochromatic wave have revealed a new phenomenon, a phase transition associated with the Landau damping regime.…”

Section: Discussionsupporting

confidence: 60%

Nonlinear density waves in the single-wave model

Marinov

Tzenov

2011

Physics of Plasmas

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The single-wave model equations are transformed to an exact hydrodynamic closure by using a class of solutions to the Vlasov equation corresponding to the waterbag model. The warm fluid dynamic equations are then manipulated by means of the renormalization group method. As a result, amplitude equations for the slowly varying wave amplitudes are derived. Since the characteristic equation for waves has in general three roots, two cases are examined. If all three roots of the characteristic equation are real, the amplitude equations for the eigenmodes represent a system of three coupled nonlinear equations. In the case, where the dispersion equation possesses one real and two complex conjugate roots, the amplitude equations take the form of two coupled equations with complex coefficients. The analytical results are then compared to the exact system dynamics obtained by solving the hydrodynamic equations numerically.

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Section: Discussionsupporting

confidence: 60%

Nonlinear density waves in the single-wave model

Marinov

Tzenov

2011

Physics of Plasmas

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“…Such nonlinear systems are relevant in many applications in hydrodynamics, 21 -24) optics, 25) oscillatory media, 26) and plasma physics. 27) It is instructive to first consider the simplest version of the system arising in thermal convection. Here nonlinearly coupled real Ginzburg-Landau equations for the local amplitudes, A and B, of two interacting families of static rolls with different orientations, are governed by 22 -24) ( )…”

Section: Introductionmentioning

confidence: 99%

Exact Solutions for Domain Walls in Coupled Complex Ginzburg–Landau Equations

Yee¹,

Tsang²,

Malomed

et al. 2011

J. Phys. Soc. Jpn.

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The complex Ginzburg Landau equation (CGLE) is a ubiquitous model for the evolution of slowly varying wave packets in nonlinear dissipative media. A front (shock) is a transient layer between a plane-wave state and a zero background.We report exact solutions for domain walls, i.e., pairs of fronts with opposite polarities, in a system of two coupled CGLEs, which describe transient layers between semi-infinite domains occupied by each component in the absence of the other one. For this purpose, a modified Hirota bilinear operator, first proposed by Bekki and Nozaki, is employed. A novel factorization procedure is applied to reduce the intermediate calculations considerably. The ensuing system of equations for the amplitudes and frequencies is solved by means of computer-assisted algebra. Exact solutions for mutually-locked front pairs of opposite polarities, with one or several free parameters, are thus generated. The signs of the cubic gain/loss, linear amplification/attenuation, and velocity of the coupled-front complex can be adjusted in a variety of configurations. Numerical simulations are performed to study the stability properties of such fronts.

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Dynamics of coupled mode solitons in bursting neural networks

2018

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Using an electrically coupled chain of Hindmarsh-Rose neural models, we analytically derived the nonlinearly coupled complex Ginzburg-Landau equations. This is realized by superimposing the lower and upper cutoff modes of wave propagation and by employing the multiple scale expansions in the semidiscrete approximation. We explore the modified Hirota method to analytically obtain the bright-bright pulse soliton solutions of our nonlinearly coupled equations. With these bright solitons as initial conditions of our numerical scheme, and knowing that electrical signals are the basis of information transfer in the nervous system, it is found that prior to collisions at the boundaries of the network, neural information is purely conveyed by bisolitons at lower cutoff mode. After collision, the bisolitons are completely annihilated and neural information is now relayed by the upper cutoff mode via the propagation of plane waves. It is also shown that the linear gain of the system is inextricably linked to the complex physiological mechanisms of ion mobility, since the speeds and spatial profiles of the coupled nerve impulses vary with the gain. A linear stability analysis performed on the coupled system mainly confirms the instability of plane waves in the neural network, with a glaring example of the transition of weak plane waves into a dark soliton and then static kinks. Numerical simulations have confirmed the annihilation phenomenon subsequent to collision in neural systems. They equally showed that the symmetry breaking of the pulse solution of the system leaves in the network static internal modes, sometime referred to as Goldstone modes.

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Ionization fronts and their velocities in a coupled Ginzburg-Landau model

Cited by 5 publications

References 21 publications

Nonlinear density waves in the single-wave model

Nonlinear density waves in the single-wave model

Exact Solutions for Domain Walls in Coupled Complex Ginzburg–Landau Equations

Dynamics of coupled mode solitons in bursting neural networks

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