Poincare covariant, the wave operators intertwine the corresponding linear and non-linear
representations of the Poincare group.
1.5.2 When the relativistic covariance is not necessarily satisfied one can use the
Hamilton-Jacobi techniques, described earlier in this introduction, cf. (1.20a). In the
context of nonlinear evolution equations, they were introduced in [14]. For the Maxwell-
Schrodinger equations, the existence of small global solutions and of modified wave oper-
ators was proved in [32]. The phase function, which defines the modification of the wave
operator, is an approximate solution of the Hamilton-Jacobi equation (even if in [32] it is
not stated that way) for a non-relativistic electron moving in an external potential.
For the Einstein vacuum equations, the existence of small global solutions (here small
means close to the Minkowski space-time) was proved in [6]. That monograph does not
study nonlinear group representation properties nor the scattering problem (except for
the property of global asymptotic flatness, relevant for differential geometry but not for
scattering theory) for the Einstein vacuum equations. Therefore, the purpose (global
existence) and the methods (generalizations of Sobolev estimates) in [6] are different from
ours. We note that, on a technical level, the Hamilton-Jacobi equation is important in [6].
In fact solutions of the Hamilton-Jacobi equation for a massless particle (i.e. the eikonal
equation) in a curved space-time are used to define norms which stay finite under the
evolution. Another point in common with the M-D equations is the existence of non-linear
constraints in the space of initial data (cf. [17] and [14]).
A soliton type of solution of the M-D equations has recently been obtained in [9], by
using variational techniques. Such solutions are important for the scattering problem with
large data.
1.6.1 NonLinear Representations of Lie Groups and Algebras. For this part the basic
reference is [10], completed by [11]. Some later references can be found in the review [30]
and the article [29] contains further developments.
1.6.1.a Let K be a Banach or Frechet space, C^(K) the space of continuous symmetric
multilinear maps Kn —• K. It is isomorphic to the space C(S™K, K) of linear continu-
ous maps from the completed (with respect to the projective tensor product topology)
symmetric tensor product of n copies of K into K. (The Hilbert-space oriented reader
is reminded that, when K is a Hilbert space, KgK corresponds to trace-class operators
on K and is smaller than the Hilbert space completion which corresponds to Hilbert-
Schmidt operators). To any fn in Cn(K) one associates the monomial fn: K K by
/»(¥) = /»(¥,...,¥),¥€#.
We denote by F{K) the space of formal series f = Y^=\ fn with product o given by
the composition of the corresponding maps Yl^Li fn fr°m K to K 1-e- if ^ = YlT hn
T(K) then the n-th term of the product is such that:
K^J7h{V) = Yl
E ^ M ® •••»#'(¥))• (1-41)
lpn ii+...+ip=n
A (formal) nonlinear representation of a Lie group G is a couple (V,K) where V is a
homomorphism G B g *-+ Vg = ]C£Li Vg OI" G m * ° t n e group of invertible elements of
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