Determinants and Their Applications in Mathematical Physics

6.8 The Kadomtsev–Petviashvili Equation 277

−

∑

r

αrA

(n)

r,n− 1

∑

s

βsA

(n)

sn

=

∑

r

∑

s

αrβs

[

AnA

(n)

rs;n− 1 ,n−

∣

∣

∣

∣

∣

A

(n)

r,n− 1 A

(n)

rn

A

(n)

s,n− 1

A

(n)

sn

∣

∣

∣

∣

∣

]

=0,

which completes the proof of the theorem.

Exercise.Prove that

logw=k+ logW,

wherekis independent ofxand, hence, thatwandW yield the same

solution of the KdV equation.

6.8 The Kadomtsev–Petviashvili Equation

6.8.1 The Non-Wronskian Solution

The KP equation is

(ut+6uux+uxxx)x+3uyy=0. (6.8.1)

The substitutionu=2vxtransforms it into

(vt+6v

2

x+vxxx)x+3vyy=0. (6.8.2)

Theorem 6.16. The KP equation in the form (6.8.2) is satisfied by the

function

v=Dx(logA),

where

A=|ars|n,

ars=δrser+

1

br+cs

,

er= exp

[

−(br+cr)x+(b

2

r−c

2

r)y+4(b

3

r+c

3

r)t+εr

]

= exp

[

−λrx+λrμry+4λr(b

2

r−brcr+c

2

r)t+εr

]

,

λr=br+cr,

μr=br−cr.

Theεrare arbitrary constants and thebrandcsare constants such that

br+cs=0, 1 ≤r, s≤n, but are otherwise arbitrary.

Proof. The proof consists of a sequence of relations similar to those

which appear in Section 6.7 on the KdV equation. Those identities which

arise from the double-sum relations (A)–(D) in Section 3.4 are as follows: