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Differential equations of infinite order are an increasingly important class of equations in theoretical physics. Such equations are ubiquitous in string field theory and have recently attracted considerable interest also from cosmologists.…
As is known, the problems for the differential equations with continuously changing order of the derivatives are not considered completely. In this paper we consider the initial and boundary value problems for this type of linear ordinary…
We consider the initial boundary value problem for free-evolution formulations of general relativity coupled to a parametrized family of coordinate conditions that includes both the moving puncture and harmonic gauges. We concentrate…
Many evolution problems in physics are described by partial differential equations on an infinite domain; therefore, one is interested in the solutions to such problems for a given initial dataset. A prominent example is the binary black…
We describe a method to construct well-posed initial value problems for not necessarily integrable equations on not necessarily simply connected quad-graphs. Although the method does not always provide a well-posed initial value problem…
The initial value problem is introduced after a thorough review of the essential geometry. The initial value equations are put into elliptic form using both conformal transformations and a treatment of the extrinsic curvature introduced…
We study the integrability of a family of birational maps obtained as reductions of the discrete Hirota equation, which are related to travelling wave solutions of the lattice KdV equation. In particular, for reductions corresponding to…
This paper provides the existence and representation of solution to an initial value problem for the general multi-term linear fractional differential equation with generalized Riemann-Liouville fractional derivatives and constant…
We describe a seemingly unnoticed feature of the text-book Maxwell-Lorentz system of classical electrodynamics which challenges its formulation in terms of an initial value problem. For point-charges, even after appropriate renormalization,…
The goal of this work is to discuss how should we impose initial values in fractional problems to ensure that they have exactly one smooth unique solution, where smooth simply means that the solution lies in a certain suitable space of…
In this article we address some issues related to the initial value problems for a rotating shallow water hyperbolic system of equations and the diffusive regularization of this system. For initial data close to the solution at rest, we…
In this letter we give fourth-order autonomous recurrence relations with two invariants, whose degree growth is cubic or exponential. These examples contradict the common belief that maps with sufficiently many invariants can have at most…
We provide a detailed treatment of relativistic Lotka-Volterra hierarchy and a kind of initial value problem with special emphasis on its the theta function representation of all algebro-geometric solutions. The basic tools involve…
We study initial value problems having dynamics ruled by discontinuous ordinary differential equations with the property of possessing a unique solution. We identify a precise class of such systems that we call solvable intitial value…
We consider the Cauchy problem for a first-order evolution equation with memory in a finite-dimensional Hilbert space when the integral term is related to the time derivative of the solution. The main problems of the approximate solution of…
In this paper we study Liouville-type properties for a class of degenerate elliptic equations driven by the fractional infinity Laplacian with nonlinear lower-order terms, \[ \Delta_\infty^{\beta}u - c\,H(u,\nabla u) - \lambda\, f(|x|,u)=0…
In contrast to regular ordinary differential equations, the problem of accurately setting initial conditions just emerges in the context of differential-algebraic equations where the dynamic degree of freedom of the system is smaller than…
All Darboux integrable difference equations on the quad-graph are described in the case of the equations that possess autonomous first-order integrals in one of the characteristics. A generalization of the discrete Liouville equation is…
The classical Liouvile integrability means that there exist $n$ independent first integrals in involution for $2n$-dimensional phase space. However, in the infinite-dimensional case, an infinite number of independent first integrals in…
Consider the diffusive HJ eq. with Dirichlet conditions, which arises in stochastic control as well as in KPZ type models of surface growth. It is known that, for $p>2$ and suitably large, smooth initial data, the sol. undergoes finite time…