Page 278 - Determinants and Their Applications in Mathematical Physics

6.7 The Korteweg–de Vries Equation 263

u rs = (τ rs ) y ,
τ rs
v rs = (τ rs ) x ,
τ rs
for all values of n and all differentiable functions f rs (x, y).
Proof.

1
(q rs ) y = (τ r+1,s ) y (τ rs ) y
τ 2
rs τ r+1,s τ rs

= τ r+1,s (τ r+1,s ) y − (τ rs ) y
τ rs
τ rs
τ r+1,s
= q rs (u r+1,s − u rs ),
which proves (a).
G 11
(u rs ) x = ,
τ 2
rs
G 13
v r+1,s − v rs = − τ r+1,s τ rs ,
G 31
u rs − u r,s−1 = ,
τ rs τ r,s−1
G 33
q rs − q r,s−1 = − .
τ rs τ r,s−1
Hence, referring to (6.2.13),

G 11 G 33
=
(q rs − q r,s−1 )(u rs ) x
q rs (u rs − u r,s−1 )(v r+1,s − v rs ) G 31 G 13
=1,
which proves (b).

6.7 The Korteweg–de Vries Equation

6.7.1 Introduction
The KdV equation is
u t +6uu x + u xxx =0. (6.7.1)
The substitution u =2v x transforms it into
2
v t +6v + v xxx =0. (6.7.2)
x
Theorem 6.13. The KdV equation in the form (6.7.2) is satisfied by the
function
v = D x (log A),