(ed.). Gravitational waves (IOP, 2001)(422s).

The linear system 263and then the four dimensional line element isds 2 = 2 −1 λ 2 dx + dx − − −1 ρ 2 (dx 2 ) 2 − (dx 3 ) 2 (14.132)where , ρ and λ depend only on x + and x − .14.5.2 The linear systemThe integrability of the nonlinear equation of motion (14.126) is reflected in theexistence of a linear system. This means that there is a set of linear differentialequations, whose compatibility conditions yield just the nonlinear equations thatone tries to solve.To formulate the linear system one must introduce a so-called spectralparameter t as an extra variable and replace v(x) by a matrix ˆv(x) which alsodepends on t.v(x 0 , x 1 ) →ˆv(x 0 , x 1 ; t). (14.133)We postulateˆv −1 ∂ µ ˆv = Q µ + 1 + t21 − t 2 P µ + 2t1 − t 2 ɛ µν P ν . (14.134)This is a generalization of v −1 ∂ µ v = Q µ + P µ , which is obtained from (14.134)in the case t = 0. (14.134) is equivalent toˆv∂ ± ˆv = Q ± + 1 ∓ t1 ± t P ±. (14.135)Here we have an integrability condition, written as∂ + ( ˆv −1 ∂ − ˆv) − ∂ − ( ˆv −1 ∂ + ˆv) + [ ˆv −1 ∂ + ˆv, ˆv −1 ∂ − ˆv] = 0 (14.136)which using (14.135) can be directly checked by calculation, making use of theintegrability condition seen before and of the equation of motion for ρ. WedefineexplicitlyEmploying (14.135) these relations become = ∂ + ( ˆv −1 ∂ − ˆv) − ∂ − ( ˆv −1 ∂ + ˆv) (14.137) = [ ˆv −1 ∂ + ˆv, ˆv −1 ∂ − ˆv]. (14.138) = ∂ + Q − − ∂ − Q + + 1 + t1 − t ∂ + P − − 1 − t1 + t ∂ − P +( ) ( )1 + t1 − t+ ∂ + P − − ∂ − P + (14.139)1 − t1 + t = [Q + , Q − ] + [P + , P − ] + 1 + t1 − t [Q +, P − ] − 1 − t1 + t [Q −, P + ]. (14.140)