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Justification of a Perturbation Approximation of the Klein–Gordon Equation

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Consider the nonlinear wave equation

uttγ2uxx+f(u) = 0

with the initial conditions

u(x,0) = εφ(x), ut(x,0) = εψ(x),

where f(u) is either of the form f(u)=c2u−σu2s+1, s=1, 2,…, or an odd smooth function with f′(0)>0 and |f′(u)|≤C02.The initial data (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; ), and prove its boundedness in xR and t>0 for all sufficiently small >0. Furthermore, we show that the error between the solution u(x, t; ) and the leading term approximation obtained by the multiple scale method is of the order 3 uniformly for xR and 0≤tT/2, as long as  is sufficiently small, T being an arbitrary positive number.
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Document Type: Original Article

Affiliations: City University of Hong Kong

Publication date: May 1, 1999

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