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152 10 Generation techniqueswhere L 1 and L 2 are again polynomial in the derivatives of ϕ A , then asolution of the differential equationsL 1 (ϕ A ,ϕ A ,a,ϕ A ,ab,...)=0, L 2 (ϕ A ,ϕ A ,a,ϕ A ,ab,...) = 0 (10.103)is also a solution of the field equations derived from L (Rund 1976). Theterm ‘variational Bäcklund transformation’ comes from the use of (10.103)for deriving Bäcklund transformations for the sine-Gordon equation andother non-linear partial differential equations. If one has one equation tobe derived from a Lagrangian L(ϕ, ϕ ,a ,...) for one function and wantsto derive an auto-Bäcklund transformation one uses the total LagrangianL tot = L(ϕ, ϕ ,a ,...) − L(ψ, ψ ,a ,...) and tries to factorize it. If successful,(10.103) provides relations between ϕ and ψ which are of lower order thanthe field equations and constitute the well-known Bäcklund transformationsfor various equations.10.10 Hirota’s methodHirota’s direct method introduces derivative operators D x defined byD n x(a ∗ b) =(∂ x − ∂ x ′) n [a(x) b(x ′ )] | x=x ′ (10.104)(Hirota 1976). Thus D x (a ∗ b) =b∂ x a − a∂ x b and D 2 x(a ∗ b) =b∂ 2 xa −2∂ x a∂ x b + a∂ 2 xb etc. This is particularly effective if the solution to theequation(s) in question can be written as a ratio f/g, say, and additionalequations can be introduced such that the resulting system can be writtenusing D operators only. It has been applied to many equations and thestandard n-soliton solutions can be found easily.10.11 Generation methods including perfect fluidsFor perfect fluids, the known generation methods are less powerful thanin the vacuum case. In this section we want to present some of the moregeneral ones. Those using only properties of a particular type of equation,e.g. a linear differential equation or a Riccati equation, for which oneknown solution may be used to get a second solution, will be treated inthe relevant chapters.10.11.1 Methods usingthe existence of KillingvectorsIf a perfect fluid solution admits a (spacelike) Killing vector ξ perpendicularto the four-velocity and has an equation of state µ = p, or admitsa (timelike) Killing vector ξ parallel to the four-velocity (rigid motion),