Page 352 - Determinants and Their Applications in Mathematical Physics

A.12 B¨acklund Transformations 337
A.12 B¨acklund Transformations

It is shown in Section 6.2.8 on brief historical notes on the Einstein and
Ernst equations that the equations
1 2 2
2 (ζ + + ζ − )∇ ζ ± =(∇ζ ± ) ,
where
2
ζ ± = φ ± ωψ (ω = −1), (A.12.1)
are equivalent to the coupled equations
2
2
2
φ∇ φ − (∇φ) +(∇ψ) =0, (A.12.2)
2
φ∇ ψ − 2∇φ · ψ = , 0 (A.12.3)
which, in turn, are equivalent to the pair
1 2 2 2 2

− φ − φ + ψ + ψ =0, (A.12.4)
ρ ρ z ρ z
φ φ ρρ + φ ρ + φ zz
∂
∂
+ ρρ z =0. (A.12.5)
ρρ ρ
∂ρ φ 2 ∂z φ 2
Given one pair of solutions of (A.12.1), it is possible to construct other
solutions by means of B¨acklund transformations.
Transformation δ
If ζ + and ζ − are solutions of (A.12.1) and

ζ = aζ − − b,
+

ζ = aζ + + b,
−
where a, b are arbitrary constants, then ζ and ζ are also solutions of
+ −
(A.12.1). The proof is elementary.
Transformation γ
If ζ + and ζ − are solution of (A.12.1) and
c

ζ = + d,
+
ζ +
c

ζ = − d,
−
ζ −

where c and d are arbitrary constants, then ζ and ζ are also solutions of

+
−
(A.12.1).
Proof.
c(ζ + + ζ − )
1 (ζ + ζ )= ,

2 + −
2ζ + ζ −