The Cauchy problem for nonlinear Klein-Gordon equations. (English) Zbl 0783.35066

Simon, Jacques C. H.; Taflin, Erik

Commun. Math. Phys. 152, No. 3, 433-478 (1993).

The nonlinear Klein-Gordon equation in \(\mathbb{R}^{n+1}\), \(n\geq 2\) is considered. It is proved that for such an equation there are “close” to zero 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 nonlinear equation.
For the manifestly Poincaré covariant equation the Poincaré-Lie algebra associated with this equation, is integrated to a nonlinear representation of the Poincaré group. This representation is linearizable by the inverse of the wave operator.

Reviewer: B.G.Konopelchenko (Novosibirsk)

MSC:

35Q53	KdV equations (Korteweg-de Vries equations)
81R05	Finite-dimensional groups and algebras motivated by physics and their representations
35S05	Pseudodifferential operators as generalizations of partial differential operators

Keywords:

Cauchy problem; global solution; nonlinear representation of the Poincaré group; nonlinear Klein-Gordon equation; asymptotic completeness

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