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We develop a diagrammatic theory for transport of waves in disordered media with weak nonlinearity. We first represent the solution of the nonlinear wave equation as a nonlinear Born series. From this, we construct nonlinear ladder and…
In this paper we address the stability of resonantly forced density waves in dense planetary rings. Already by Goldreich & Tremaine (1978) it has been argued that density waves might be unstable, depending on the relationship between the…
Q-conditional symmetries (nonclassical symmetries) for a general class of two-component reaction-diffusion systems with constant diffusivities are studied. Using the recently introduced notion of Q-conditional symmetries of the first type…
The global existence of bounded solutions to reaction-diffusion systems with fractional diffusion in the whole space $\mathbb R^N$ is investigated. The systems are assumed to preserve the non-negativity of initial data and to dissipate…
Nonlinear dynamical systems possessing reflection symmetry have an invariant subspace in the phase space. The dynamics within the invariant subspace can be random or chaotic. As a system parameter changes, the motion transverse to the…
The global existence of bounded weak solutions to a diffusion system modeling biofilm growth is proven. The equations consist of a reaction-diffusion equation for the substrate concentration and a fourth-order Cahn-Hilliard-type equation…
The understanding of the large-scale structure formation requires the resolution of coupled nonlinear equations describing the cosmic density and velocity fields. This is a complicated problem that, for the last decade, has been essentially…
Nonequilibrium phenomena of the phase transitions are studied. It is shown that due to finite relaxation time of the particle distributions, the use of scalar background dependent distribution functions is inconsistent.This observation may…
We present Wideband Back-Projection Diffusion, an end-to-end probabilistic framework for approximating the posterior distribution induced by the inverse scattering map from wideband scattering data. This framework produces highly accurate…
Mass-conserving reaction-diffusion (MCRD) systems are widely used to model phase separation and pattern formation in cell polarity, biomolecular condensates, and ecological systems. Numerical simulations and formal asymptotic analysis…
We study the longitudinal motion of beam particles under the action of a single resonator wave induced by the beam itself. Based on the method of multiple scales we derive a system of coupled amplitude equations for the slowly varying part…
In this paper, we study unique, globally defined uniformly bounded weak solutions for a class of semilinear reaction-diffusion-advection systems. The coefficients of the differential operators and the initial data are only required to be…
This paper aims to study existence condition of possible bursting oscillations generated by low frequency excitation of a nonlinear vibratory system in the presence of parametric excitation. Slow-fast dissection technique and numerical…
Building up on classical linear formulations, we posit that a broad class of problems in signal synthesis and in signal recovery are reducible to the basic task of finding a point in a closed convex subset of a Hilbert space that satisfies…
Density perturbations and their dynamic evolution from early to late times can be used for an improved understanding of interesting physical phenomena both in cosmology and in the context of heavy-ion collisions. We discuss the spectrum and…
A mathematical model describing the initial stage of the capture into autoresonance for nonlinear oscillating systems with combined parametric and external excitation is considered. The solutions with unboundedly growing amplitude and…
Sulci are surface folds commonly seen in strained soft elastomers and form via a strongly subsubcriticalcritical, yet scale-free instability. Treating the threshold for nonlinear instability as a nonlinear critical point, we explain the…
In this Series, we study the weakly nonlinear dynamics of chemically active particles near the threshold for spontaneous motion. In this part, we focus on steady solutions and develop an `adjoint method' for deriving the nonlinear amplitude…
Pattern formation in uniaxial polymeric liquid crystals is studied for different dynamic closure approximations. Using the principles of mesoscopic non-equilibrium thermodynamics in a mean-field approach, we derive a Fokker-Planck equation…
Fully developed turbulence is a universal and scale-invariant chaotic state characterized by an energy cascade from large to small scales where the cascade is eventually arrested by dissipation. In this article, we show how to harness these…