Modulational instability analysis of the Peregrine soliton

We perform modulational instability analysis of the Peregrine soliton. The eigensystem of the linearized perturbations results in time-dependent gain curve. The instantaneous stability of the Peregrine soliton is studied at different times and in terms of modulated spacial width. A correlation between the most unstable eigenmode and the time evolution of the Peregrine soliton is established. Our analysis explains the bifurcation of the Peregrine soliton into Ma breathers and the generation of shock waves. The theoretical approach and numerical procedure followed here may be applied to any other localized solution with nontrivial time dependence.     

Structural instability of a soliton for the Davey-Stewartson II equation

An algebraic soliton of the Davey-Stewartson II equation looses structural stability under a small perturbation of the initial data. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 118, No. 3, pp. 354–361, March, 1999.     

Critical temperature of soliton state stability in quasi-one-dimensional systems

The temperature effect on soliton state formation in quasi-one-dimensional systems is considered in the self-consistent field approximation. The critical temperature above which soliton states are nonstable is shown to exist. The crirical temperature dependence on nonlinearity parameter and system dimension is found. Soliton density at which the critical temperature is maximal is calculated.     

Fingering instability in particle systems

In many multiphase systems, material interfaces can be destabilized by shocks. Small disturbances at these interfaces can grow in size to form large-scale fingers. We consider a shock propagating through a system that consists of two types of particles, of different mass, that are initially separated by an interface, but are free to mix. In the classical case of immiscible fluids, the finger of heavy fluid propagating into the light fluid grows faster and becomes much thinner than the finger of light fluid propagating into the heavy fluid. We show that collisions between particles of different types lead to shock focusing that causes a secondary flow that is initially similar to the fluid case. However, the particle system can exhibit completely different qualitative behavior in the nonlinear-growth phase and can give rise to the situation where the finger of heavy material is actually wider than the finger of the light material. We show that this qualitative change is due to a strong decompression that occurs in the heavy material. We also show that microscopic mixing can have an important impact on finger growth.     

Resonant instability in multidimensional Lyapunov systems

The stability of a point of rest of a Lyapunov system, which describes the perturbed motion of a dynamical system with an arbitrary number of degrees of freedom, is investigated. It is assumed that the characteristic equation of the system of the first approximation has a number of imaginary roots, while the quadratic part of the integral is not sign-definite. The case of third-order resonance is considered, when the stability problem is solved by the first non-linear (quadratic) terms.     

Hamiltonian and super-Hamiltonian systems of a hierarchy of soliton equations

By considering an isospectral eigenvalue problem, a hierarchy of soliton equations are derived. Two types of extensions are presented by enlarging the associated spectral problem. With the aid of generalized trace identity and the super-trace identity, the Hamiltonian and super-Hamiltonian structures for the integrable extensions are established.     

Complexity and stability of soliton management in periodically modulated and random systems

A brief introduction is given to the concept of the soliton management, i.e., stable motion of localized pulses in media with strong periodic (or, sometimes, random) inhomogeneity, or conditions for the survival of solitons in models with strong time‐periodic modulation of linear or nonlinear coefficients. It is demonstrated that a class of systems can be identified, in which solitons remain robust inherently coherent objects in seemingly “hostile” environments. Most physical models belonging to this class originate in nonlinear optics and Bose‐Einstein condensation, although other examples are known too (in particular, in hydrodynamics). In this paper, the complexity of the soliton‐management systems, and the robustness of solitons in them are illustrated using a recently explored fiber‐optic setting combining a periodic concatenation of nonlinear and dispersive segments (the split‐step model) for bimodal optical signals (i.e., ones with two polarizations of light), which includes the polarization mode dispersion, i.e., random linear mixing of the two polarization components at junctions between the fiber segment. © 2008 Wiley Periodicals, Inc. Complexity, 2008     

Chaos-based communication systems in non-ideal channels

Recently, many chaos-based communication systems have been proposed. They can present the many interesting properties of spread spectrum modulations. Besides, they can represent a low-cost increase in security. However, their major drawback is to have a Bit Error Rate (BER) general performance worse than their conventional counterparts. In this paper, we review some innovative techniques that can be used to make chaos-based communication systems attain lower levels of BER in non-ideal environments. In particular, we succinctly describe techniques to counter the effects of finite bandwidth, additive noise and delay in the communication channel. Although much research is necessary for chaos-based communication competing with conventional techniques, the presented results are auspicious.     

Some problems of instability of difference systems

Siberian Mathematical Journal -     

Observation of two soliton propagation in an erbium doped inhomogeneous lossy fiber with phase modulation

We consider optical pulse propagation in an Erbium doped inhomogeneous lossy optical fiber with time dependent phase modulation, which is governed by a system of Generalized Inhomogeneous Nonlinear Schrödinger Maxwell–Bloch (GINLS–MB) equation. Multi-soliton propagation is studied analytically by means of deriving associated Lax pair and the soliton solutions are obtained using Darboux transformation. By suitably adjusting the group velocity dispersion and nonlinearity parameter, we discuss various soliton dynamics such as periodic distributed amplification, pulse compression etc. In each case, we demonstrate the influence of inhomogeneous parameter. Finally we investigate the pulse compression through nonlinear tunneling.     

Cross-diffusion induced instability and stability in reaction-diffusion systems

In a reaction-diffusion system, diffusion can induce the instability of a uniform equilibrium  which is stable with respect to a constant perturbation, as shown by Turing in 1950s. We show that cross-diffusion can destabilize  a uniform equilibrium  which is stable for the kinetic and self-diffusion reaction systems; on the other hand, cross-diffusion can also stabilize  a uniform equilibrium which is stable for the kinetic system but unstable for the self-diffusion reaction system. Application is given to predator-prey system with preytaxis and vegetation pattern formation in a water-limited ecosystem.     

Indexing domains of instability for Hamiltonian systems

Based upon the understanding of the global topologies of the singular subset, its complement, and the hyperbolic subset in the symplectic group, in this paper we study the domains of instability for hyperbolic Hamiltonian systems and define a characteristic index for such domains. This index is defined via the Maslov-type index theory for symplectic paths starting from the identity defined by C. Conley, E. Zehnder, and Y. Long, and the hyperbolic index of symplectic matrices. The old problem of the relation between the non-degenerate local minimality and the instability of hyperbolic extremal loops in the calculus of variation is also studied via this new index for the domains of instability. Received July 4, 1997     

Resonance and marginal instability of switching systems

We analyze the so-called Marginal Instability of linear switching systems, both in continuous and discrete time. This is a phenomenon of unboundedness of trajectories when the Lyapunov exponent is zero. We disprove two recent conjectures of Chitour, Mason and Sigalotti (2012) stating that for generic systems, the resonance is sufficient for marginal instability and for polynomial growth of the trajectories. The concept of resonance originated with the same authors is modified. A characterization of marginal instability under some mild assumptions on the system is provided. These assumptions can be verified algorithmically and are believed to be generic. Finally, we analyze possible types of fastest asymptotic growth of trajectories. An example of a marginally unstable pair of matrices with non-polynomial growth is given.     

Modulation instability of solutions of the nonlinear Schrödinger equation

Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 84, No. 2, pp. 163–172, August, 1990.     

Optimal level of diagnostics in distributed communication systems

A unit (multiplexor, switch, terminal, processor, etc.) at a remote site is inspected occasionally from a central site to see if it is functioning properly. Two stochastic models are proposed for monitoring remote units whose operational status can be described by an alternating 0–1 process. Optimal inspection schedules are derived based on the objective of maximizing the overall steady state system availability while limiting inspections per-unit-time costs and customer inconvenience.