Page 354 - Determinants and Their Applications in Mathematical Physics

A.12 B¨acklund Transformations 339

∇ψ 1 2
∇· = (φ∇ ψ − φ · ψ), (A.12.10)
φ φ 2

∇ψ 1 2
∇· = (φ∇ ψ − 2∇φ · ψ), (A.12.11)
φ 2 φ 3
1 2 2
2
∇ (log φ)= [φ∇ φ − (∇φ) ], (A.12.12)
φ 2
2
∇ (log ρ)=0. (A.12.13)
Applying (A.12.12) and (A.12.11), the coupled equations (A.12.2) and
(A.12.3) become
2
2
2
φ ∇ (log φ)+(∇ψ) =0, (A.12.14)

∇ψ
∇· =0. (A.12.15)
φ 2
Transformation β (Ehlers)
If the pair P(φ, ψ) is a solution of (A.12.4) and (A.12.5), and φ and ψ are

functions which satisfy the relations
ρ

a. φ = ,
φ
∂ψ ωρ ∂ψ
b. = − ,
∂ρ φ 2 ∂z
∂ψ ωρ ∂ψ 2
c. = ,(ω = −1),
∂z φ 2 ∂ρ
then the pair P (φ ,ψ ) is also a solution.

Proof. Applying (A.12.6) and (A.12.7) to (A.12.15),

1 ∂ψ 1 ∂ψ
∇· , =0,
2
2
φ ∂ρ φ ∂z

∂ ρ ∂ψ ∂ ρ ∂ψ
+ =0,
2
∂ρ φ ∂ρ ∂z φ ∂z
2
which is satisfied by (b) and (c). Eliminating ψ from (b) and (c),
2
2
∂ φ ∂ψ ∂ φ ∂ψ
+ =0,
∂ρ ρ ∂ρ ∂z ρ ∂z
2
2
∂ ψ 1 ∂ψ ∂ ψ 2 ∂φ ∂ψ ∂φ ∂ψ
− + = − + .
∂ρ 2 ρ ∂ρ ∂z 2 φ ∂ρ ∂ρ ∂z ∂z
Hence, referring to (A.12.8) and (a),
2φ 1 ρ ∂φ
∂ψ ρ ∂φ ∂ψ
2

∇ ψ = − −
2
2
ρ φ φ ∂ρ ∂ρ φ ∂z ∂z
ρ
2φ ∂
∂ψ ∂
∂ψ
ρ
= +
ρ ∂ρ φ ∂ρ ∂z φ ∂z