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

Wronskianwdefined as follows:

w=

∣

∣Dj−^1

x

(ψi)

∣

∣

n

,

where

ψi= exp

(

1

4

b

2

iy

)

φi,

φi=pie

1 / 2

i

+qie

− 1 / 2

i

,

ei= exp(−bix+b

3

i

t)

andbi,pi, andqiare arbitrary functions ofi.

The proof is obtained by replacingzbyyin the proof of the first line of

(6.7.60) withF= 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 puttingy=0.

6.9 The Benjamin–Ono Equation.................

6.9.1 Introduction

The notationω

2

=−1 is used in this section, asiandjare indispensable

as row and column parameters.

Theorem.The Benjamin–Ono equation in the form

AxA

∗

x−

1

2

[

A

∗

(Axx+ωAt)+A(Axx+ωAt)

∗

]

=0, (6.9.1)

whereA

∗

is the complex conjugate ofA, is satisfied for all values ofnby

the determinant

A=|aij|n,

where

aij=

{

2 ci

ci−cj

,j=i

1+ωθi,j=i

(6.9.2)

θi=cix−c

2

it−λi, (6.9.3)

and where theciare distinct but otherwise arbitrary constants and theλi

are arbitrary constants.

The proof which follows is a modified version of the one given by

Matsuno. It begins with the definitions of three determinantsB,P, andQ.