Numerical Study of Nonlinear Equations with Infinite Number of Derivatives

Abstract

We study equations with infinitely many derivatives. Equations of this type form a new class of equations in mathematical physics. These equations originally appeared in p-adic and later in fermionic string theories and their investigation is of much interest in mathematical physics and applications, in particular in cosmology. Differential equation with infinite number of derivatives could be written as nonlinear integral equations. We perform numerical investigation of solutions of the equations. It is established that these equations have two different regimes of the solutions: interpolating and periodic. The critical value of the parameter q separating these regimes is found to be q^2=1.37. Convergence of iterative procedure for these equations is proved.