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The cauchy problem for non-linear Klein-Gordon equations

Jacques SimonErik Taflin

subject

Cauchy problemPure mathematicsMathematical analysisHilbert spaceStatistical and Nonlinear Physicssymbols.namesakeNorm (mathematics)Poincaré groupLie algebrasymbolsTrivial representationCovariant transformationKlein–Gordon equationMathematical PhysicsMathematics

description

We consider in ℝ n+1,n≧2, the non-linear Klein-Gordon equation. We prove for such an equation that there is a neighbourhood of zero in a Hilbert space of initial conditions for which the Cauchy problem has global solutions and on which there is asymptotic completeness. The inverse of the wave operator linearizes the non-linear equation. If, moreover, the equation is manifestly Poincare covariant then the non-linear representation of the Poincare Lie algebra, associated with the non-linear Klein-Gordon equation is integrated to a non-linear representation of the Poincare group on an invariant neighbourhood of zero in the Hilbert space. This representation is linearized by the inverse of the wave operator. The Hilbert space is, in both cases, the closure of the space of the differentiable vectors for the linear representation of the Poincare group, associated with the Klein-Gordon equation, with respect to a norm defined by the representation of the enveloping algebra.

yearjournalcountryeditionlanguage
1993-03-01Communications in Mathematical Physics
https://doi.org/10.1007/bf02096615