Related papers: New Results on Compact Structures
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…
The paper presents methods of eigenvalue localisation of regular matrix polynomials, in particular, stability of matrix polynomials is investigated. For this aim a stronger notion of hyperstability is introduced and widely discussed. Matrix…
We investigate stationary, spatially localized patterns in lattice dynamical systems that exhibit bistability. The profiles associated with these patterns have a long plateau where the pattern resembles one of the bistable states, while the…
We consider a non-linear system modelling the dynamics of a linearly elastic body immersed in an incompressible viscous fluid, without damping on the elastic part. We prove local existence of strong solutions and global existence and…
The formation of self-organized patterns and localized states are ubiquitous in Nature. Localized states containing trivial symmetries such as stripes, hexagons, or squares have been profusely studied. Disordered patterns with non-trivial…
We present a series of recent results on the well-posedness of very singular parabolic stochastic partial differential equations. These equations are such that the question of what it even means to be a solution is highly non-trivial. This…
Various methods of constructing solvable few-body models are reviewed, with an emphasis on direct interactions with few degrees of freedom, as an alternative to the use of local quantum field theories. Several applications are discussed.
In this paper we study the set of rational solutions of equations defined by power sums symmetric polynomials with coefficients in a finite field. We do this by means of applying a methodology which relies on the study of the geometry of…
We consider a prototypical model in which a nonlinear field (continuum or discrete) evolves on a flexible substrate which feeds back to the evolution of the main field. We identify the underlying physics and potential applications of such a…
Bilinear systems of equations are defined, motivated and analyzed for solvability. Elementary structure is mentioned and it is shown that all solutions may be obtained as rank one completions of a linear matrix polynomial derived from…
In this chapter we review recent results concerning localized and extended dissipative solutions of the discrete complex Ginzburg-Landau equation. In particular, we discuss discrete diffraction effects arising both from linear and nonlinear…
We investigate the periodic and stationary solutions of distribution-dependent stochastic differential equations. While generally, the semigroups associated with the equations are nonlinear, we show that the methods of weak convergence and…
We extend the definition of $n$-dimensional difference equations to complex order $\alpha\in \mathbb{C} $. We investigate the stability of linear systems defined by an $n$-dimensional matrix $A$ and derive conditions for the stability of…
We investigate the existence of spatially localised solutions, in the form of discrete breathers, in general damped and driven nonlinear lattice systems of coupled oscillators. Conditions for the exponential decay of the difference between…
Partial differential equations with discrete (concentrated) state-dependent delays are studied. The existence and uniqueness of solutions with initial data from a wider linear space is proven first and then a subset of the space of…
Discrete models have a long tradition in engineering, including finite state machines, Boolean networks, Petri nets, and agent-based models. Of particular importance is the question of how the model structure constrains its dynamics. This…
This paper presents a novel multi-scale method for elliptic partial differential equations with arbitrarily rough coefficients. In the spirit of numerical homogenization, the method constructs problem-adapted ansatz spaces with uniform…
We present the applications of methods from nonlinear local harmonic analysis for calculations in nonlinear collective dynamics described by different forms of Vlasov-Maxwell-Poisson equations. Our approach is based on methods provided the…
The existence and analyticity of solutions to linear systems of moment differential equations with analytic coefficients is studied. The relation of solutions of such systems with respect to linear moment differential equations is…
This study presents a method for constructing a sequence of approximate solutions of increasing accuracy to general equilibrium models on nonlocal domains. The method is based on a technique originated from dynamical systems theory. The…