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

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
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