Page 256 - 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 internal waves in a stratified fluid of great
depth and in the propagation of nonlinear Rossby waves in a rotating fluid.
It can be expressed in the form
u t +4uu x + H{u xx } =0,
where H{f(x)} denotes the Hilbert transform of f(x) defined as

1 , ∞ f(y)
H{f(x)} = P dy
π y − x
−
and where P denotes 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
2A x A = A (A xx + ωA t )+ A(A xx + ωA t ) ∗ (ω = −1),
∗
∗
x
∗
where A is the complex conjugate of A, and found a solution in which
A is a determinant of arbitrary order whose diagonal elements are linear
in x and t and 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
∂ ∂P −1
∂ ∂P −1
ρ P + ρ P =0,
∂ρ ∂ρ ∂z ∂z
where the matrix P is defined as

1 1 ψ
P = 2 2 . (6.2.6)
φ ψ φ + ψ
φ is the gravitational potential and is real and ψ is 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.