Nonlinear Sciences

The classical theory of modulation instability (MI) attributed to Bespalov-Talanov in optics and Benjamin-Feir for water waves is just a linear approximation of nonlinear effects and has limitations that have been corrected using the exact weakly nonlinear theory of wave propagation. We report results of experiments in both, optics and hydrodynamics, which are in excellent agreement with nonlinear theory. These observations clearly demonstrate that MI has wider band of unstable frequencies than predicted by the linear stability analysis. The range of areas where the nonlinear theory of MI can be applied is actually much larger than considered here.

We describe, for the first time, the full 2D scattering of long-lived breathers in a model hexagonal lattice of atoms. The chosen system, representing an idealized model of mica, combines a Lennard-Jones interatomic potential with an "egg-box" harmonic potential well surface. We investigate the dependence of breather properties on the ratio of the well depths associated to the interaction and on-site potentials. High values of this ratio lead to large spatial displacements in adjacent chains of atoms and thus enhance the two dimensional character of the quasi-one-dimensional breather solutions. This effect is further investigated during breather-breather collisions by following the constrained energy density function in time for a set of randomly excited mobile breather solutions. Certain collisions lead to 60 ∘ scattering, and collisions of mobile and stationary breathers can generate a rich variety of states.

From infiltration of water into the soil during rainstorms to seasonal plant growth and death, the ecohydrological processes that are thought to be relevant to the formation of banded vegetation patterns in drylands occur across multiple timescales. We propose a new fast-slow switching model in order to capture these processes on appropriate timescales within a conceptual modeling framework based on reaction-advection-diffusion equations. The fast system captures hydrological processes that occur on minute to hour timescales during and shortly after major rainstorms, assuming a fixed vegetation distribution. These include key feedbacks between vegetation biomass and downhill surface water transport, as well as between biomass and infiltration rate. The slow system acts between rain events, on a timescale of days to months, and evolves vegetation and soil moisture. Modeling processes at the appropriate timescales allows parameter values to be set by the actual processes they capture. This reduces the number of parameters that are chosen expressly to fit pattern characteristics, or to artificially slow down fast processes by the orders of magnitude required to align their timescales with the biomass dynamics. We explore the fast-slow switching model through numerical simulation on a one-dimensional hillslope, and find agreement with certain observations about the pattern formation phenomenon, including band spacing and upslope colonization rates. We also find that the predicted soil moisture dynamics are consistent with time series data that has been collected at a banded vegetation site. This fast-slow model framework introduces a tool for investigating the possible impact of changes to frequency and intensity of rain events in dryland ecosystems.

The study of nonlinear waves that collapse in finite time is a theme of universal interest, e.g. within optical, atomic, plasma physics, and nonlinear dynamics. Here we revisit the quintessential example of the nonlinear Schrodinger equation and systematically derive a normal form for the emergence of blowup solutions from stationary ones. While this is an extensively studied problem, such a normal form, based on the methodology of asymptotics beyond all algebraic orders, unifies both the dimension-dependent and power-law-dependent bifurcations previously studied; it yields excellent agreement with numerics in both leading and higher-order effects; it is applicable to both infinite and finite domains; and it is valid in all (subcritical, critical and supercritical) regimes.

This article describes the conversion of the two-dimensional Primordial Particle System into a threedimensional model that exhibits comparable features. We present the transformed model here in the form of a pseudocode implementation and detail the modifications required for this conversion.

Dark solitons are common topological excitations in a wide array of nonlinear waves. The dark soliton excitation energy, crucial for exploring dark soliton dynamics, is necessarily calculated in a renormalized form due to its existence on a finite background. Despite its tremendous importance and success, the renormalized energy form was firstly only suggested with no detailed derivation, and was then "derived" in the grand canonical ensemble. In this work, we revisit this fundamental problem and provide an alternative and intuitive derivation of the energy form from the fundamental field energy by utilizing a limiting procedure that conserves number of particles. Our derivation yields the same result, putting therefore the dark soliton energy form on a solid basis.

A discrete and periodic complex Ginzburg-Landau equation, coupled to a discrete mean equation, is systematically derived from a driven and dissipative oscillator model, close to the onset of a supercritical Hopf bifurcation. The oscillator model is inspired by recent experiments exploring active vibrations of quasi-one-dimensional lattices of self-propelled millimetric droplets bouncing on a vertically vibrating fluid bath. Our systematic derivation provides a direct link between the constitutive properties of the lattice system and the coefficients of the resultant amplitude equations, paving the way to compare the emergent nonlinear dynamics---namely discrete bright and dark solitons, breathers, and traveling waves---against experiments. Further, the amplitude equations allow us to rationalize the successive bifurcations leading to these distinct dynamical states. The framework presented herein is expected to be applicable to a wider class of oscillators characterized by the presence of a dynamic coupling potential between particles. More broadly, our results point to deeper connections between nonlinear oscillators and the physics of active and driven matter.

We study a (1+1)-dimensional field theory based on (ψlnψ ) 2 potential. There are three degenerate minima at ψ=0 and ψ=±1 . There are novel, asymmetric kink solutions of the form ψ=∓exp(−exp(±x)) connecting the minima at ψ=0 and ψ=∓1 . The domains with ψ=0 repel the linear excitations, the waves (e.g. phonons). Topology restricts the domain sequences and therefore the ordering of the domain walls. Collisions between domain walls are rich for properties such as transmission of kinks and particle conversion, etc. To our knowledge this is the first example of kinks with super-exponential profiles and super-exponential tails. Finally, we provide a comparison of these results with the ϕ 6 model and its half-kink solution.

A nonlinear reaction-diffusion system with cross-diffusion describing the COVID-19 outbreak is studied using the Lie symmetry method. A complete Lie symmetry classification is derived and it is shown that the system with correctly-specified parameters admits highly nontrivial Lie symmetry operators, which do not occur for all known reaction-diffusion systems. The symmetries obtained are also applied for finding exact solutions of the system in the most interesting case from applicability point of view. It is shown that the exact solutions derived possess all necessary properties for describing the pandemic spread under 1D approximation in space and lead to the distributions, which qualitatively correspond to the measured data of the COVID-19 spread in Ukraine.

In many oscillatory or excitable systems, dynamical patterns emerge which are stationary or periodic up in a moving frame of reference. Examples include traveling waves or spiral waves in chemical systems or cardiac tissue. We present a unified theoretical framework for the drift of such patterns under small external perturbations, in terms of overlap integrals between the perturbation and the adjoint critical eigenfunctions of the linearised operator (i.e. `response functions'). For spiral waves, the finite radius of the spiral tip trajectory as well as spiral wave meander are taken into account. Different coordinates systems can be chosen, depending on whether one wants to predict the motion of the spiral wave tip, the time-averaged tip path, or the center of the meander flower. The framework is applied to analyse the drift of a meandering spiral wave in a constant external field in different regimes.