The initial value problem for ordinary differential equations with infinitely many derivatives

@article{Grka2012TheIV,
  title={The initial value problem for ordinary differential equations with infinitely many derivatives},
  author={Przemyslaw G{\'o}rka and Humberto Prado and Enrique G. Reyes},
  journal={Classical and Quantum Gravity},
  year={2012},
  volume={29}
}
We study existence, uniqueness and regularity of solutions for ordinary differential equations with infinitely many derivatives such as (linearized versions of) nonlocal field equations of motion appearing in particle physics, nonlocal cosmology and string theory. We develop a Lorentzian functional calculus via Laplace transform which allows us to interpret rigorously operators of the form f(∂t) on the half line, in which f is an analytic function. We find the most general solution to the… 

On equations with infinitely many derivatives: integral transforms and the Cauchy problem

We analyze initial value problems for ordinary differential equations with infinitely many derivatives such as (linearized versions of) nonlocal field equations of motion appearing in particle

Differential Equations with Infinitely Many Derivatives and the Borel Transform

Differential equations with infinitely many derivatives, sometimes also referred to as “nonlocal” differential equations, appear frequently in branches of modern physics such as string theory,

On linear differential equations with infinitely many derivatives

Differential equations with infinitely many derivatives, sometimes also referred to as ``nonlocal'' differential equations, appear frequently in branches of modern physics such as string theory,

On a General Class of Nonlocal Equations

Motivated by recent developments in cosmology and string theory, we introduce a functional calculus appropriate for the study of non-linear nonlocal equations of the form f(Δ)u = U(x, u(x)) on

On a General Class of Nonlocal Equations

Motivated by recent developments in cosmology and string theory, we introduce a functional calculus appropriate for the study of non-linear nonlocal equations of the form f(Δ)u = U(x, u(x)) on

Infinite-derivative linearized gravity in convolutional form

This article aims to transform the infinite-order Lagrangian density for ghost-free infinite-derivative linearized gravity into non-local form. To achieve it, we use the theory of generalized

The Borel transform and linear nonlocal equations: applications to zeta-nonlocal field models

We define rigorously operators of the form $f(\partial_t)$, in which $f$ is an analytic function on a simply connected domain. Our formalism is based on the Borel transform on entire functions of

Non-local scalar fields in static spacetimes via heat kernels

We solve the non-local equation −e−` 2 φ = J for i) static scalar fields in static spacetimes and ii) time-dependent scalar fields in ultrastatic spacetimes. Corresponding equations are rewritten as

Infinite Derivative Gravity: A finite number of predictions

Ghost-free Infinite Derivative Gravity (IDG) is a modifed gravity theory which can avoid the singularities predicted by General Relativity. This thesis examines the effect of IDG on four areas of

Ghost-free infinite-derivative dilaton gravity in two dimensions

We present the ghost-free infinite-derivative extensions of the Spherically-Reduced Gravity (SRG) and Callan-Giddings-Harvey-Strominger (CGHS) theories in two space-time dimensions. For the case of

References

SHOWING 1-10 OF 41 REFERENCES

Dynamics with infinitely many derivatives: The Initial value problem

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

Dynamics with infinitely many derivatives: variable coefficient equations

Infinite order differential equations have come to play an increasingly significant role in theoretical physics. Field theories with infinitely many derivatives are ubiquitous in string field theory

Nonlinear Dynamics Equation in p-Adic String Theory

We investigate nonlinear pseudodifferential equations with infinitely many derivatives. These are equations of a new class, and they originally appeared in p-adic string theory. Their investigation

Functional calculus via laplace transform and equations with infinitely many derivatives

We study nonlocal linear equations of the form f(∂t)ϕ=J(t), t≥0, in which f is an entire function. We develop an appropriate functional calculus via Laplace transform, we solve the aforementioned

The equation of the $ p$-adic open string for the scalar tachyon field

We study the structure of solutions of the one-dimensional non-linear pseudodifferential equation describing the dynamics of the -adic open string for the scalar tachyon field . We explain the role

A new formulation of the initial value problem for nonlocal theories

The Problem of Nonlocality in String Theory

Vector-valued Laplace Transforms and Cauchy Problems

This monograph gives a systematic account of the theory of vector-valued Laplace transforms, ranging from representation theory to Tauberian theorems. In parallel, the theory of linear Cauchy

Differential operators of infinite order with real arguments and their applications

Part 1 Preliminaries: Convolution of Distributions Fourier Transforms of Distributions Entire Functions of Exponential Type that are Bounded on Rn The Dirichlet Kernels Markov Type Theorems The

Zeta-nonlocal scalar fields

AbstractWe consider some nonlocal and nonpolynomial scalar field models originating from p-adic string theory. An infinite number of space-time derivatives is determined by the operator-valued