Justification of a Perturbation Approximation of the Klein–Gordon Equation

Abstract:

Consider the nonlinear wave equation

with the initial conditions , where f(u). is either of the form f(u) = c²u − σu^2s+1, s = 1, 2, …, or an odd smooth function with f′(0) > 0 and . The initial data φ(x) ∈ C² and ψ(x) ∈ C¹ are odd periodic functions that have the same period. We establish the global existence and uniqueness of the solution u(x, t; ε), and prove its boundedness in x ∈ ℝ and t > 0 for all sufficiently small ε > 0. Furthermore, we show that the error between the solution u(x, t; ε) and the leading term approximation obtained by the multiple scale method is of the order ε³ uniformly for x ∈ ℝ and 0 ≤ t ≤ T/ ε², as long as ε is sufficiently small, T being an arbitrary positive number.