Page 296 - Determinants and Their Applications in Mathematical Physics

6.9 The Benjamin–Ono Equation 281

Theorem. The KP equation in the form (6.8.2) is satisfied by the
Wronskian w defined as follows:

j−1
w = D (ψ i ) ,

x
n
where
1 2
ψ i = exp b y φ i ,
4 i
1/2 −1/2
φ i = p i e + q i e ,
i i
3
e i = exp(−b i x + b t)
i
and b i , p i , and q i are arbitrary functions of i.
The proof is obtained by replacing z by y in the proof of the first line of
(6.7.60) with F = 0 in the KdV section. The reverse procedure is invalid. If
the KP equation is solved first, it is not possible to solve the KdV equation
by putting y =0.

6.9 The Benjamin–Ono Equation

6.9.1 Introduction
2
The notation ω = −1 is used in this section, as i and j are indispensable
as row and column parameters.
Theorem. The Benjamin–Ono equation in the form

∗ 1 ∗ ∗
A x A − A (A xx + ωA t )+ A(A xx + ωA t ) =0, (6.9.1)
2
x
where A is the complex conjugate of A, is satisfied for all values of n by
∗
the determinant
A = |a ij | n ,
where

2c i , j = i
a ij = c i −c j (6.9.2)
1+ ωθ i ,j = i
2
θ i = c i x − c t − λ i , (6.9.3)
i
and where the c i are distinct but otherwise arbitrary constants and the λ i
are arbitrary constants.
The proof which follows is a modified version of the one given by
Matsuno. It begins with the definitions of three determinants B, P, and Q.