Related papers: An extended variational theory for nonlinear evolu…
We consider the evolution of populations under the joint action of mutation and differential reproduction, or selection. The population is modelled as a finite-type Markov branching process in continuous time, and the associated…
This paper is concerned with the theory of generic non-normal nonlinear evolutionary equations, with potential applications in Fluid Dynamics and Optics. Two theoretical models are presented. The first is a model two-level non-normal…
Density perturbations and their dynamic evolution from early to late times can be used for an improved understanding of interesting physical phenomena both in cosmology and in the context of heavy-ion collisions. We discuss the spectrum and…
This is a complete study of the dynamics of polynomial planar vector fields whose linear part is a multiple of the identity and whose nonlinear part is a contracting homogeneous polynomial. The contracting nonlinearity provides the…
Space time fractional nonlinear evolution equations have been widely applied for describing various types of physical mechanism of natural phenomena in mathematical physics and engineering. The proposed generalized exp expansion method…
We review the construction of homological evolutionary vector fields on infinite jet spaces and partial differential equations. We describe the applications of this concept in three tightly inter-related domains: the variational Poisson…
In this paper we construct nonlinear partial differential equations in more than 3 independent variables, possessing a manifold of analytic solutions with high, but not full, dimensionality. For this reason we call them ``partially…
The multiphase Whitham modulation equations with $N$ phases have $2N$ characteristics which may be of hyperbolic or elliptic type. In this paper a nonlinear theory is developed for coalescence, where two characteristics change from…
We develop a noncommutative invariant theory for ordinary linear differential operators on Riemann surfaces. For a monic binomially normalized operator $L=\sum_{k=0}^n {n\choose k}a_kD^{\,n-k}$, $a_0=1$, with coefficients in an associative…
A variational framework is defined for vertical slice models with three dimensional velocity depending only on x and z. The models that result from this framework are Hamiltonian, and have a Kelvin-Noether circulation theorem that results…
New manifestly gauge-invariant forms of two-dimensional generalized dispersive long-wave and Nizhnik-Veselov-Novikov systems of integrable nonlinear equations are presented. It is shown how in different gauges from such forms famous…
The process of evolutionary diversification unfolds in a vast genotypic space of potential outcomes. During the past century there have been remarkable advances in the development of theory for this diversification, and the theory's success…
Convolutional Neural Networks define an exceptionally powerful class of models, but are still limited by the lack of ability to be spatially invariant to the input data in a computationally and parameter efficient manner. In this work we…
We prove that any potential symmetry of a system of evolution equations reduces to a Lie symmetry through a nonlocal transformation of variables. Based on this fact is our method of group classification of potential symmetries of systems of…
We construct nonlinear extensions of Dirac's relativistic electron equation that preserve its other desirable properties such as locality, separability, conservation of probability and Poincar\'e invariance. We determine the constraints…
We study the theory of systems with constraints from the point of view of the formal theory of partial differential equations. For finite-dimensional systems we show that the Dirac algorithm completes the equations of motion to an…
A potential representation for the subset of traveling solutions of nonlinear dispersive evolution equations is introduced. The procedure involves a reduction of a third order partial differential equation to a first order ordinary…
In this paper we first introduce an innovative equivalent norm in the Musielak-Orlicz Sobolev spaces in a very general setting and we then present a new result on the boundedness of the solutions of a wide class of nonlinear Neumann…
A classification of all possible realizations of the Galilei, Galilei-similitude and Schroedinger Lie algebras in three-dimensional space-time in terms of vector fields under the action of the group of local diffeomorphisms of the space…
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…