Related papers: Snaking and isolas of localised states in bistable…

We prove the existence of time-periodic solutions consisting of patterns built up from two states, one with small amplitude and the other one with large amplitude, in general nonlinear Hamiltonian finite-size lattices with global coupling.…

We study self-trapped localized nonlinear states in the form of truncated Bloch waves in one-dimensional optical lattices, which appear in the gaps of the linear bandgap spectrum. We demonstrate the existence of families of such localized…

In one-dimensional Hermitian tight-binding models, mobility edges separating extended and localized states can appear in the presence of properly engineered quasi-periodical potentials and coupling constants. On the other hand, mobility…

We address the existence and stability of localized modes in the two-dimensional (2D) linear Schroedinger lattice with two symmetric nonlinear sites embedded into it, and a generalization for moderately localized nonlinearity featuring two…

Recent work on the behaviour of localised states in pattern forming partial differential equations has focused on the traditional model Swift-Hohenberg equation which, as a result of its simplicity, has additional structure --- it is…

We introduce a system with one or two amplified nonlinear sites ("hot spots", HSs) embedded into a two-dimensional linear lossy lattice. The system describes an array of evanescently coupled optical or plasmonic waveguides, with gain…

We analyze the existence and stability of two kinds of self-trapped spatially localized gap modes, gap solitons and truncated nonlinear Bloch waves, in one-and two-dimensional optical or matter-wave media with self-focusing nonlinearity,…

We present a general approach to prove the existence, both locally and globally in amplitude, of fully localised multi-dimensional patterns in partial differential equations containing a compact spatial heterogeneity. While one-dimensional…

We theoretically investigate the dynamics, bifurcation structure and stability of dark localized states emerging in Kerr cavities in the presence of second- and fourth-order dispersion. These states form through the locking of uniform wave…

Many biological systems, such as metabolic pathways, exhibit bistability behavior: these biological systems exhibit two distinct stable states with switching between the two stable states controlled by certain conditions. Since…

In this paper, we study a nonlocal evolution system. We apply abstract results from the bifurcation theory to obtain the existence of coexistence states. Their stability are investigated as well.

We consider scalar lattice differential equations posed on square lattices in two space dimensions. Under certain natural conditions we show that wave-like solutions exist when obstacles (characterized by "holes") are present in the…

This paper studies a two microbial species model in competition for a single resource in the chemostat including general interspecific density-dependent growth rates with distinct removal rates for each species. We give the necessary and…

We prove the existence of exponentially localised and time-periodic solutions in general nonlinear Hamiltonian lattice systems. Like normal modes, these localised solutions are characterised by collective oscillations at the lattice sites…

Many engineering structures are composed of weakly coupled sectors assembled in a cyclic and ideally symmetric configuration, which can be simplified as forced Duffing oscillators. In this paper, we study the emergence of localized states…

We study quantum systems on a discrete bounded lattice (lattice billiards). The statistical properties of their spectra show universal features related to the regular or chaotic character of their classical continuum counterparts. However,…

We study nonlinear excitations described by DNLS-type equations with so-called competing nonlinearities. These are the nonlinearities that consist of two power terms with coefficients of different sign. A key feature of these models is the…

The modified discrete nonlinear Schr\"odinger equation is used to study the formation of stationary localized states in a one-dimensional lattice with a single impurity and an asymmetric dimer impurity. A periodically modulated and a…

Networks of limit cycle oscillators can show intricate patterns of synchronization such as splay states and cluster synchronization. Here we analyze dynamical states that display a continuum of seemingly independent splay clusters. Each…

We study the conditions under which species interaction, as described by continuous versions of the competitive Lotka-Volterra model (namely the nonlocal Kolmogorov-Fisher model, and its differential approximation), can support the…