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Solitons and breathers are nonlinear modes that exist in a wide range of physical systems. They are fundamental solutions of a number of nonlinear wave evolution equations, including the uni-directional nonlinear Schr\"odinger equation…
The Kruskal-Segur approach to selection theory in diffusion-limited or Laplacian growth is extended via combination with the Zauderer decomposition scheme. This way nonlinear bulk equations become tractable. To demonstrate the method, we…
Inspired by one--dimensional light--particle systems, the dynamics of a non-Hamiltonian system with long--range forces is investigated. While the molecular dynamics does not reach an equilibrium state, it may be approximated in the…
As an introductory lecture to the workshop an overview is given over continuum models for homoepitaxial surface growth using partial differential equations (PDEs). Their {\em heuristic derivation} makes use of inherent symmetries in the…
The evolution of an initial perturbation in Vlasov plasma is studied in the intrinsically nonlinear long-time limit dominated by the effects of particle trapping. After the possible transient linear exponential Landau damping, the evolution…
In relativistic cosmology, the formation of nonlinear inhomogeneities can induce non-negligible backreaction on late-time expansion. Among the important consequences for precision cosmology is the potential impact on the linear growth of…
A first-principles theory is developed for the general evolution of a key structural characteristic of planar granular systems - the cell order distribution. The dynamic equations are constructed and solved in closed form for a number of…
We study the existence and uniqueness of the solution of a non-linear coupled system constituted of a degenerate diffusion-growth-fragmentation equation and a differential equation, resulting from the modeling of bacterial growth in a…
The statistical effects of different conditions in physical vapor deposition, such as sputter deposition, have on thin film morphology has long been the subject of interest. One notable effect is that of column development due to…
We investigate numerically the influence of an homogeneous shear flow on the spinodal decomposition of a binary mixture by solving the Cahn-Hilliard equation in a two-dimensional geometry. Several aspects of this much studied problem are…
In this review we try to capture some of the recent excitement induced by experimental developments, but also by a large volume of theoretical and computational studies addressing multi-component nonlinear Schrodinger models and the…
The fragmentation of filaments in molecular clouds has attracted a lot of attention as there seems to be a relation between the evolution of filaments and star formation. The study of the fragmentation process has been motivated by simple…
As a first step to understand anomalous kinetic roughening with multifractality in recent experiments of the vapor deposition polymerization (VDP) growth, we study a simple toy model of the VDP growth in a (1+1)-dimensional lattice, along…
Diffusion Limited Aggregation (DLA) has served for forty years as a paradigmatic example for the creation of fractal growth patterns. In spite of thousands of references no exact result for the fractal dimension $D$ of DLA is known. In this…
We write the correlation function of dark matter particles, xi(r), as the sum of two terms - one which accounts for nonlinear evolution, and dominates on small scales, and another which is essentially the term from linear theory, and…
We discuss the late time evolution of the gravitational clustering in an expanding universe, based on the nonlinear scaling relations (NSR) which connect the nonlinear and linear two point correlation functions. The existence of critical…
We introduce two discrete models of a collection of colliding particles with stored momentum and study the asymptotic growth of the mean-square displacement of an active particle. We prove that the models are superdiffusive in one dimension…
Stochastic growth phenomena on curved interfaces are studied by means of stochastic partial differential equations. These are derived as counterparts of linear planar equations on a curved geometry after a reparametrization invariance…
We present a kinetic theory for the evolution of the phase-space distribution of dark matter particles in galaxy halos in the presence of a cosmological spectrum of fluctuations. This theory introduces a new way to model the formation and…
Mathematical models describing the spatial spreading and invasion of populations of biological cells are often developed in a continuum modelling framework using reaction-diffusion equations. While continuum models based on linear diffusion…