Determinants and Their Applications in Mathematical Physics

A.12 B ̈acklund Transformations 339

∇·

(

∇ψ

φ

)

=

1

φ^2

(φ∇

2

ψ−∇φ·∇ψ), (A.12.10)

∇·

(

∇ψ

φ

2

)

=

1

φ

3

(φ∇

2

ψ− 2 ∇φ·∇ψ), (A.12.11)

∇

2

(logφ)=

1

φ

2

[φ∇

2

φ−(∇φ)

2

], (A.12.12)

∇

2

(logρ)=0. (A.12.13)

Applying (A.12.12) and (A.12.11), the coupled equations (A.12.2) and

(A.12.3) become

φ

2

∇

2

(logφ)+(∇ψ)

2

=0, (A.12.14)

∇·

(

∇ψ

φ

2

)

=0. (A.12.15)

Transformationβ(Ehlers)

If the pairP(φ, ψ) is a solution of (A.12.4) and (A.12.5), andφ

′

andψ

′

are

functions which satisfy the relations

a.φ

′

=

ρ

φ

,

b.

∂ψ

′

∂ρ

=−

ωρ

φ

2

∂ψ

∂z

,

c.

∂ψ

′

∂z

=

ωρ

φ

2

∂ψ

∂ρ

,(ω

2

=−1),

then the pairP

′

(φ

′

,ψ

′

) is also a solution.

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

∇·

(

1

φ

2

∂ψ

∂ρ

,

1

φ

2

∂ψ

∂z

)

=0,

∂

∂ρ

(

ρ

φ

2

∂ψ

∂ρ

)

+

∂

∂z

(

ρ

φ

2

∂ψ

∂z

)

=0,

which is satisfied by (b) and (c). Eliminatingψfrom (b) and (c),

∂

∂ρ

(

φ

2

ρ

∂ψ

′

∂ρ

)

+

∂

∂z

(

φ

2

ρ

∂ψ

′

∂z

)

=0,

∂

2

ψ

′

∂ρ

2

−

1

ρ

∂ψ

′

∂ρ

+

∂

2

ψ

′

∂z

2

=−

2

φ

(

∂φ

∂ρ

∂ψ

′

∂ρ

+

∂φ

∂z

∂ψ

′

∂z

)

.

Hence, referring to (A.12.8) and (a),

∇

2

ψ

′

=

2 φ

ρ

[(

1

φ

−

ρ

φ

2

∂φ

∂ρ

)

∂ψ

′

∂ρ

−

ρ

φ

2

∂φ

∂z

∂ψ

′

∂z

]

=

2 φ

ρ

[

∂

∂ρ

(

ρ

φ

)

∂ψ

′

∂ρ

+

∂

∂z

(

ρ

φ

)

∂ψ

′

∂z

]