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

(ζ++ζ−)∇

2

ζ±=(∇ζ±)

2

,

where

ζ±=φ±ωψ (ω

2

=−1), (A.12.1)

are equivalent to the coupled equations

φ∇

2

φ−(∇φ)

2

+(∇ψ)

2

=0, (A.12.2)

φ∇

2

ψ− 2 ∇φ·∇ψ=, 0 (A.12.3)

which, in turn, are equivalent to the pair

φ

(

φρρ+

1

ρ

φρ+φzz

)

−φ

2

ρ−φ

2

z+ψ

2

ρ+ψ

2

z=0, (A.12.4)

∂

∂ρ

(

ρψρ

φ

2

)

+

∂

∂z

(

ρψz

φ

2

)

=0. (A.12.5)

Given one pair of solutions of (A.12.1), it is possible to construct other

solutions by means of B ̈acklund transformations.

Transformationδ

Ifζ+andζ−are solutions of (A.12.1) and

ζ

′

+=aζ−−b,

ζ

′

−

=aζ++b,

wherea, bare arbitrary constants, thenζ

′

+

andζ

′

−

are also solutions of

(A.12.1). The proof is elementary.

Transformationγ

Ifζ+andζ−are solution of (A.12.1) and

ζ

′

+

=

c

ζ+

+d,

ζ

′

−

=

c

ζ−

−d,

wherecanddare arbitrary constants, thenζ

′

+andζ

′

−are also solutions of

(A.12.1).

Proof.

1

2

(ζ

′

++ζ

′

−)=

c(ζ++ζ−)

2 ζ+ζ−

,