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.


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
Full Text: DOI


[1] Flato, M., Pinczon, G., Simon, J.C.H.: Non-linear representations of Lie groups. Ann. Sci. Ec. Norm. Super.10, 405–418 (1977) · Zbl 0384.22005
[2] Hörmander, L.: Remarks on the Klein-Gordon equation. Journée Equations aux dérivées partielles. Saint-Jean de Monts 1987, Soc. Math. France (1987)
[3] Hörmander, L.: The analysis of linear partial differential operators, Vol. III. Berlin Heidelberg, New York: Springer 1985 · Zbl 0601.35001
[4] Klainerman, S.: Global existence of small amplitude solutions to nonlinear Klein-Gordon equations in four space-time dimensions. Commun. Pure Appl. Math.38, 631–641 (1985) · Zbl 0597.35100
[5] Shatah, J.: Normal forms and quadratic non-linear Klein-Gordon equations. Commun. Pure Appl. Math.38, 685–696 (1985) · Zbl 0597.35101
[6] Simon, J.C.H.: A wave operator for a non-linear Klein-Gordon equation. Lett. Math. Phys.7, 387–398 (1983) · Zbl 0539.35007
[7] Simon, J.C.H., Taflin, E.: Wave operators and analytic solutions for systems of non-linear Klein-Gordon equations and non-linear Schrödinger equations. Commun. Math. Phys.99, 541–562 (1985) · Zbl 0615.47034
[8] Warner, G.: Harmonic analysis on semi-simple Lie groups I, Vol. 188. Berlin, Heidelberg, New York: Springer 1972 · Zbl 0265.22020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.