Related papers: The Nikolaevskiy equation with dispersion
We develop a complete stability theory for two-dimensional periodic traveling waves of reaction-diffusion systems. More precisely, we identify a diffusive spectral stability assumption, prove that it implies nonlinear stability and provide…
Recently, a concept of deterministic and stochastic turbulence has been introduced based on experiments with a boundary layer. In these experiments, the flow was driven with controlled random perturbation; in addition, natural ambient noise…
Wave turbulence and eddy turbulence are the two regimes that we may encounter in nature. The attention of fluid mechanics being mainly focused on incompressible hydrodynamics, it is usually the second regime that is treated in books,…
We provide several characterizations of convergence to unstable equilibria in nonlinear systems. Our current contribution is three-fold. First we present simple algebraic conditions for establishing local convergence of non-trivial…
We study wave turbulence in systems with two special properties: a large number of fields (large $N$) and a nonlinear interaction that is strongly local in momentum space. The first property allows us to find the kinetic equation at all…
This contribution investigates an original stochastic approach for the emergence of stop-and-go waves in traffic flow, a collective phenomenon with significant safety and environmental implications. Using a stable nonlinear car-following…
A class of dynamical models of turbulence living on a one-dimensional dyadic-tree structure is introduced and studied. The models are obtained as a natural generalization of the popular GOY shell model of turbulence. These models are found…
The evolution of the amplitude of two nonlinearly interacting waves is considered, via a set of coupled nonlinear Schroedinger-type equations. The dynamical profile is determined by the wave dispersion laws (i.e. the group velocities and…
The Gross-Pitaevskii equation (GPE) plays an important role in the description of Bose-Einstein condensates (BECs) at the mean-field level. The GPE belongs to the class of non-linear Schr\"odinger equations which are known to feature…
Uchaikin suggested a mathematical model of an anomalous diffusion in a space was suggested. This model origins in an investigation of processes in complex systems with variable structure: glasses, liquid crystals, biopolymers, proteins and…
Wave turbulence is by nature a multiple time scale problem for which there is a natural asymptotic closure. The main result of this analytical theory is the kinetic equation that describes the long-time statistical behaviour of such…
Determination of stability and instability of singular points in nonlinear dynamical systems is an important issue that has attracted considerable attention in different fields of engineering and science. So far, different well-defined…
Statistical systems with time-periodic spatially non-uniform forces are of immense importance in several areas of physics. In this paper, we provide an analytical expression of the time-periodic probability distribution function of…
Periodic travelling waves (PTW) are a common solution type of partial differential equations. Such models exhibit multistability of PTWs, typically visualised through the Busse balloon, and parameter changes typically lead to a cascade of…
This paper provides a new unified framework for second-moment stability of discrete-time linear systems with stochastic dynamics. Relations of notions of second-moment stability are studied for the systems with general stochastic dynamics,…
Modelling sediment transport in environmental turbulent fluids is a challenge. This article develops a sound model of the lateral transport of suspended sediment in environmental fluid flows such as floods and tsunamis. The model is…
The procedure of comprehensive analysis of instability of current sheathes in a wide range of frequencies and wave lengths in the electrically neutral approximation has been developed. This comprehensive analysis of instability is based on…
We address stability of a class of Markovian discrete-time stochastic hybrid systems. This class of systems is characterized by the state-space of the system being partitioned into a safe or target set and its exterior, and the dynamics of…
The wave turbulence equation is an effective kinetic equation that describes the dynamics of wave spectrum in weakly nonlinear and dispersive media. Such a kinetic model has been derived by physicists in the sixties, though the…
Generalized Navier-Stokes equations which were proposed recently to describe active turbulence in living fluids are analyzed rigorously. Results on wellposedness and stability in the $L^2(\mathbb{R}^n)$-setting are derived. Due to the…