Determinants and Their Applications in Mathematical Physics

6.7 The Korteweg–de Vries Equation 263

urs=

(τrs)y

τrs

,

vrs=

(τrs)x

τrs

,

for all values ofnand all differentiable functionsfrs(x, y).

Proof.

(qrs)y=

1

τ

2

rs

∣

∣

∣

∣

(τr+1,s)y (τrs)y

τr+1,s τrs

∣

∣

∣

∣

=

τr+1,s

τrs

[

(τr+1,s)y

τr+1,s

−

(τrs)y

τrs

]

=qrs(ur+1,s−urs),

which proves (a).

(urs)x=

G 11

τ

2

rs

,

vr+1,s−vrs=−

G 13

τr+1,sτrs

,

urs−ur,s− 1 =

G 31

τrsτr,s− 1

,

qrs−qr,s− 1 =−

G 33

τrsτr,s− 1

Hence, referring to (6.2.13),

(qrs−qr,s− 1 )(urs)x

qrs(urs−ur,s− 1 )(vr+1,s−vrs)

=

G 11 G 33

G 31 G 13

=1,

which proves (b).

6.7 The Korteweg–de Vries Equation

6.7.1 Introduction

The KdV equation is

ut+6uux+uxxx=0. (6.7.1)

The substitutionu=2vxtransforms it into

vt+6v

2

x

+vxxx=0. (6.7.2)

Theorem 6.13. The KdV equation in the form (6.7.2) is satisfied by the

function

v=Dx(logA),