Page 292 - Determinants and Their Applications in Mathematical Physics

6.8 The Kadomtsev–Petviashvili Equation 277
(n) (n)
− α r A β s A
r,n−1 sn
r s

(n)

(n) A r,n−1 (n)
= α r β s A n A A rn
rs;n−1,n − (n) (n)
A s,n−1
r s A sn
=0,
which completes the proof of the theorem.
Exercise. Prove that
log w = k + log W,
where k is independent of x and, hence, that w and W 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

(u t +6uu x + u xxx ) x +3u yy =0. (6.8.1)
The substitution u =2v x transforms it into
2
(v t +6v + v xxx ) x +3v yy =0. (6.8.2)
x
Theorem 6.16. The KP equation in the form (6.8.2) is satisfied by the
function
v = D x (log A),
where

A = |a rs | n ,
1
a rs = δ rs e r + ,

2 2 3 3
b r + c s
e r = exp −(b r + c r )x +(b − c )y +4(b + c )t + ε r
r r r r
2 2
= exp −λ r x + λ r µ r y +4λ r (b − b r c r + c )t + ε r ,
r r
λ r = b r + c r ,
µ r = b r − c r .
The ε r are arbitrary constants and the b r and c s are constants such that
b r + c s =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: