Determinants and Their Applications in Mathematical Physics

6.2 Brief Historical Notes 241

linear operators and a determinant of arbitrary order whose elements are

defined as integrals.

The verifications given in Sections 6.7 and 6.8 of the non-Wronskian

solutions of both the KdV and KP equations apply purely determinantal

methods and are essentially those published by Vein and Dale in 1987.

6.2.7 The Benjamin–Ono Equation

The Benjamin–Ono (BO) equation is a nonlinear integro-differential equa-

tion which arises in the theory of internalwaves in astratified fluid of great

depth and in the propagation of nonlinear Rossbywaves in arotating fluid.

It can be expressed in the form

ut+4uux+H{uxx}=0,

whereH{f(x)}denotes the Hilbert transform off(x) defined as

H{f(x)}=

1

π

P

∫

∞

−∞

f(y)

y−x

dy

and wherePdenotes the principal value.

In a paper published in 1988, Matsuno introduced a complex substitution

into the BO equation which transformed it into a more manageable form,

namely

2 AxA

∗

x=A

∗

(Axx+ωAt)+A(Axx+ωAt)

∗

(ω

2

=−1),

whereA

∗

is the complex conjugate ofA, and found a solution in which

Ais a determinant of arbitrary order whose diagonal elements are linear

inxandtand whose nondiagonal elements contain a sequence of distinct

arbitrary constants.

6.2.8 The Einstein and Ernst Equations

In the particular case in which a relativistic gravitational field is axially

symmetric, the Einstein equations can be expressed in the form

∂

∂ρ

(

ρ

∂P

∂ρ

P

− 1

)

+

∂

∂z

(

ρ

∂P

∂z

P

− 1

)

=0,

where the matrixPis defined as

P=

1

φ

[

1 ψ

ψφ

2

+ψ

2

]

. (6.2.6)

φis the gravitational potential and is real andψis either real, in which

case it is the twist potential, or it is purely imaginary, in which case it has

no physical significance. (ρ, z) are cylindrical polar coordinates, the angular

coordinate being absent as the system is axially symmetric.