Nonlinear Equations with Infinitely many Derivatives

Abstract

We study the generalized bosonic string equation

Delta e(-c) (Delta)phi = U(x, phi), c > 0

on Euclidean space R(n). First, we interpret the nonlocal operator Delta e(-c) (Delta) using entire vectors of Delta in L(2)(R(n)), and we show that if U(x, phi) = phi(x) + f (x), in which f is an element of L(2)(R(n)), then there exists a unique real-analytic solution to the Euclidean bosonic string in a Hilbert space H(c,) (infinity) (R(n)) we define precisely below. Second, we consider the case in which the potential U(x, phi) in the generalized bosonic string equation depends nonlinearly on phi, and we show that this equation admits real-analytic solutions in H(c,infinity)(R(n)) under some symmetry and growth assumptions on U. Finally, we show that the above given equation admits real-analytic solutions in H(c,infinity)(R(n)) if U(x, phi) is suitably near U(0)(x, phi) = phi, even if no symmetry assumptions are imposed.