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Recurrence plots exhibit line structures which represent typical behaviour of the investigated system. The local slope of these line structures is connected with a specific transformation of the time scales of different segments of the…
The paper investigates localized deformation patterns resulting from the onset of instabilities in lattice structures. The study is motivated by previous observations on discrete hexagonal lattices, where the onset of non-uniform,…
We construct spatiotemporal localized envelope solutions of a (3+1)-dimensional nonlinear Schr\"{o}dinger equation with varying coefficients such as dispersion, nonlinearity and gain parameters through similarity transformation technique.…
It is well-known that localized topological defects (solitons) experience recoil when they suffer an impact by incident particles. Higher-dimensional topological defects develop distinctive wave patterns propagating along their worldvolume…
We demonstrate the existence of a large number of exact solutions of plane Couette flow, which share the topology of known periodic solutions but are localized in space. Solutions of different size are organized in a snakes-and-ladders…
Motivated by numerical continuation studies of coupled mechanical oscillators, we investigate branches of localized time-periodic solutions of one-dimensional chains of coupled oscillators. We focus on Ginzburg--Landau equations with…
The present paper deals with the wave propagation in a particular two dimensional structure, obtained from a localized perturbation of a reference periodic medium. This reference medium is a ladder like domain, namely a thin periodic…
Recently it has been discovered that some nonlinear evolution equations in 2+1 dimensions, which are integrable by the use of the Spectral Transform, admit localized (in the space) soliton solutions. This article briefly reviews some of the…
We investigated the effect of time delays on phase configurations in a set of two-dimensional coupled phase oscillators. Each oscillator is allowed to interact with its neighbors located within a finite radius, which serves as a control…
The mode-locking regions of a dynamical system are the subsets of the parameter space of the system within which there exists an attracting periodic solution. For piecewise-linear continuous maps, these regions have a curious chain…
Localized patterns are spatially confined structures that arise in lattice dynamical systems and play an important role in physics, biology, and materials science. While their existence and bifurcation structure are well-understood, the…
We study the shape of convective rolls in the Marine Atmospheric Boundary Layer from Synthetic Aperture Radar images of the ocean. We propose a multiscale analysis with structure functions which allow an easy generalization to analyse…
In dimensions greater than or equal to 3, the local structure of a singular holomorphic foliation conceals a globally defined foliation on the projective space of dimension one less. In this paper, we will discuss how the global dynamics of…
We study the linearization of a class of thick K-branes, namely, four-dimensional domain walls generated by a scalar field with particular nonstandard kinetic terms. The master equations for linear perturbations are derived from the point…
Many natural shapes have most of their characterizing features concentrated over a few regions in space. For example, humans and animals have distinctive head shapes, while inorganic objects like chairs and airplanes are made of…
We study localised exact coherent structures in plane Poiseuille flow that are relative periodic orbits. They are obtained from extended states in smaller, periodically continued domains, by increasing the length to obtain streamwise…
I investigate models with scalar fields in 5 dimensions that exhibit thick-brane configurations with a non-trivial metric. I show that an appropriate coupling to the scalar curvature allows for periodic configurations, which, however, are…
In more than one spatial dimension, resonant linear conversion from one wave type to another can have a more complex geometry than the familiar 'avoided crossing' of one-dimensional problems. In previous work we have shown that helical ray…
Branched rough paths, used to solve ODEs on $\mathbb{R}$, have been generalised in two different directions. In one direction, there are regularity structures aimed at solving SPDEs on $\mathbb{R}$. In the other direction, there are…
We develop the theory of nonlinear localised modes (intrinsic localised modes or discrete breathers) in two-dimensional (2D) photonic crystal waveguides. We consider different geometries of the waveguides created by an array of nonlinear…