Related papers: Do All Integrable Evolution Equations Have the Pai…
The path integral approach to the quantization of one degree-of-freedom Newtonian particles is considered within the discrete time-slicing approach, as in Feynman's original development. In the time-slicing approximation the quantum…
We show the continuous dependence of solutions of linear nonautonomous second order parabolic partial differential equations (PDEs) with bounded delay on coefficients and delay. The assumptions are very weak: only convergence in the weak-*…
We review some surprising links which have been discovered in the last few years between the theory of certain ordinary differential equations, and particular integrable lattice models and quantum field theories in two dimensions. An…
In several cases of nonlinear dispersive PDEs, the difference between the nonlinear and linear evolutions with the same initial data, i.e. the integral term in Duhamel's formula, exhibits improved regularity. This property is usually called…
We introduce and analyse a continuum model for an interacting particle system of Vicsek type. The model is given by a non-linear kinetic partial differential equation (PDE) describing the time-evolution of the density $f_t$, in the single…
In this paper some open problems for Painlev\'e equations are discussed. In particular the following open problems are described: (i) the Painlev\'e equivalence problem; (ii) notation for solutions of the Painlev\'e equations; (iii)…
We describe a variant of the dressing method giving alternative representation of multidimensional nonlinear PDE as a system of Integro-Differential Equations (IDEs) for spectral and dressing functions. In particular, it becomes single…
In this article, using DiPerna-Lions theory \cite{Di-Li}, we investigate linear second order stochastic partial differential equations with unbounded and degenerate non-smooth coefficients, and obtain several conditions for existence and…
In recent years, progress toward the classification of superintegrable systems with higher order integrals of motion has been made. In particular, a complete classification of all exotic potentials with a third or a fourth order integrals,…
We present a unified convergence theory for gradient-based training of neural network methods for partial differential equations (PDEs), covering both physics-informed neural networks (PINNs) and the Deep Ritz method. For linear PDEs, we…
A new class of nonlinear partial differential equations with distributed in space and time state-dependent delay is investigated. We find appropriate assumptions on the kernel function which represents the state-dependent delay and discuss…
Partial Differential Equations (PDEs) describe phenomena ranging from turbulence and epidemics to quantum mechanics and financial markets. Despite recent advances in computational science, solving such PDEs for real-world applications…
In this paper we introduce and investigate a new kind of functional (including ordinary and evolutionary partial) differential equations. The main goal of this paper is to explore our new philosophy by some examples on functional ODEs and…
We study the Laurent property, the irreducibility and co-primeness of discrete integrable and non-integrable equations. First we study a discrete integrable equation related to the Somos-4 sequence, and also a non-integrable equation as a…
We discuss the dimensional characterization of the solutions space of a formally integrable system of partial differential equations and provide certain formulas for calculations of these dimensional quantities.
We propose a discrete form for an equation due to Gambier and which belongs to the class of the fifty second order equations that possess the Painleve property. In the continuous case, the solutions of the Gambier equation is obtained…
In this article we present a way of treating stochastic partial differential equations with multiplicative noise by rewriting them as stochastically perturbed evolutionary equations in the sense of \cite{picardbook}, where a general…
This article is focused on two related topics within the study of partial differential equations (PDEs) that illustrate a beautiful connection between dynamics, topology, and analysis: stability and spatial dynamics. The first is a property…
A wide class of non-autonomous nonlinear parabolic partial differential equations with delay is studied. We allow in our investigations different types of delays such as constant, time-dependent, state-dependent (both discrete and…
We study a family of Li\'enard--type equations. Such equations are used for the description of various processes in physics, mechanics and biology and also appear as traveling--wave reductions of some nonlinear partial differential…