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Register a new userPropagation of instability fronts in modulationally unstable systems
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We study evolution of pulses propagating through focusing nonlinear media. Small disturbance on the smooth initial non-uniform background leads to formation of the region of strong nonlinear oscillations. We develop here an asymptotic method for finding the law of motion of the front of this region. The method is applied to the focusing nonlinear Schroedinger equation for the particular cases of Talanov and Akhmanov-Sukhorukov-Khokhlov initial distributions with zero initial phase. The approximate analytical results agree well with the exact numerical solutions for these two problems.
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Evidence is presented of universal behavior in modulationally unstable media. An ensemble of nonlinear evolution equations, including three partial differential equations, an integro-differential equation, a nonlocal system and a differential-difference equation, is studied. Collectively, these systems arise in a variety of applications in the physical and mathematical sciences, including water waves, optics, acoustics, Bose-Einstein condensation, and more. All these models exhibit modulational instability, namely, the property that a constant background is unstable to long-wavelength perturbations. In this work, each of these systems is studied analytically and numerically for a number of different initial perturbations of the constant background, and it is shown that, for all systems and for all initial conditions considered, the dynamics gives rise to a remarkably similar structure comprised of two outer, quiescent sectors separated by a wedge-shaped central region characterized by modulated periodic oscillations. A heuristic criterion that allows one to compute some of the properties of the central oscillation region is also given.
Non-equilibrium dissipative systems usually exhibit multistability, leading to the presence of propagative domain between steady states. We investigate the front propagation into an unstable state in discrete media. Based on a paradigmatic model of coupled chain of oscillators and populations dynamics, we calculate analytically the average speed of these fronts and characterize numerically the oscillatory front propagation. We reveal that different parts of the front oscillate with the same frequency but with different amplitude. To describe this latter phenomenon we generalize the notion of the Peierls-Nabarro potential, achieving an effective continuous description of the discreteness effect.
We undertake a systematic exploration of recurrent patterns in a 1-dimensional Kuramoto-Sivashinsky system. For a small, but already rather turbulent system, the long-time dynamics takes place on a low-dimensional invariant manifold. A set of equilibria offers a coarse geometrical partition of this manifold. A variational method enables us to determine numerically a large number of unstable spatiotemporally periodic solutions. The attracting set appears surprisingly thin - its backbone are several Smale horseshoe repellers, well approximated by intrinsic local 1-dimensional return maps, each with an approximate symbolic dynamics. The dynamics appears decomposable into chaotic dynamics within such local repellers, interspersed by rapid jumps between them.
A periodically inhomogeneous Schrodinger equation is considered. The inhomogeneity is reflected through a non-uniform coefficient of the linear and non-linear term in the equation. Due to the periodic inhomogeneity of the linear term, the system may admit spectral bands. When the oscillation frequency of a localized solution resides in one of the finite band gaps, the solution is a gap soliton, characterized by the presence of infinitely many zeros in the spatial profile of the soliton. Recently, how to construct such gap solitons through a composite phase portrait is shown. By exploiting the phase-space method and combining it with the application of a topological argument, it is shown that the instability of a gap soliton can be described by the phase portrait of the solution. Surface gap solitons at the interface between a periodic inhomogeneous and a homogeneous medium are also discussed. Numerical calculations are presented accompanying the analytical results.
The empirical velocity of a reaction-diffusion front, propagating into an unstable state, fluctuates because of the shot noises of the reactions and diffusion. Under certain conditions these fluctuations can be described as a diffusion process in the reference frame moving with the average velocity of the front. Here we address pushed fronts, where the front velocity in the deterministic limit is affected by higher-order reactions and is therefore larger than the linear spread velocity. For a subclass of these fronts -- strongly pushed fronts -- the effective diffusion constant $D_fsim 1/N$ of the front can be calculated, in the leading order, via a perturbation theory in $1/N ll 1$, where $Ngg 1$ is the typical number of particles in the transition region. This perturbation theory, however, overestimates the contribution of a few fast particles in the leading edge of the front. We suggest a more consistent calculation by introducing a spatial integration cutoff at a distance beyond which the average number of particles is of order 1. This leads to a non-perturbative correction to $D_f$ which even becomes dominant close to the transition point between the strongly and weakly pushed fronts. At the transition point we obtain a logarithmic correction to the $1/N$ scaling of $D_f$. We also uncover another, and quite surprising, effect of the fast particles in the leading edge of the front. Because of these particles, the position fluctuations of the front can be described as a diffusion process only on very long time intervals with a duration $Delta t gg tau_N$, where $tau_N$ scales as $N$. At intermediate times the position fluctuations of the front are anomalously large and non-diffusive. Our extensive Monte-Carlo simulations of a particular reacting lattice gas model support these conclusions.
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