Nonlinear Equations with Infinitely Many Derivatives

We study the generalized bosonic string equation on Euclidean space ?n. First, we interpret the nonlocal operator ?e-c ? using entire vectors of ? in L2(?n), and we show that if U(x, ?) = ?(x) + f(x), in which f ? L2(?n), then there exists a unique real-analytic solution to the Euclidean bosonic string in a Hilbert space Hc, ? (?n) we define precisely below. Second, we consider the case in which the potential U(x, ?) in the generalized bosonic string equation depends nonlinearly on ?, and we show that this equation admits real-analytic solutions in Hc, ? (?n) under some symmetry and growth assumptions on U. Finally, we show that the above given equation admits real-analytic solutions in Hc, ? (?n) if U(x, ?) is suitably near U0(x, ?), even if no symmetry assumptions are imposed. © 2009 Birkhäuser Verlag Basel/Switzerland.