Integrability of Nonlinear Systems

Nonlinear Waves, Solitons, and IST 9
Finally, it is worth remarking upon the solution of the linear equation (1.17)
with u(x, 0) = f(x) given, and decaying sufficiently just as |x| →∞. The solution
is obtained by Fourier transforms and is found to be
u(x, t) = 1 ∫ ∞
b 0 (k)e i(kx+k3t) dk, (1.24)
2π
∫ ∞
−α
where b 0 (k) = f(x)e −ikx dx =û(k, 0), and where û(k, t) denotes the Fourier
transform at any time t. While the general solution establishes existence,
−∞
qualitative information can be obtained by further study. Asymptotic analysis as
t →∞(stationary phase-steepest descent methods) establishes that the solution
decays as
( )
1 b0 (|z|)+b 0 (−|z|)
u(x, t) ∼
Ai(z)
(3t) 1/3 2
+ 1 ( )
(1.25)
b0 (|z|) − b 0 (−|z|)
Ai ′ (z),
(3t) 2/3 2i|z|
where z = x/(3t) 1/3 and Ai(z) is the Airy function,
∫ ∞
(
exp i
Ai(z) = 1
2π
−∞
(sz + s3
3
))
ds, (1.26)
(cf. [3] for further details).
The method of Fourier transforms generalizes readily, e.g., it is applicable
to any evolutionary PDE with constant coefficients. The scheme for solving a
linear problem with a dispersion relation ω(k), e.g., (1.17) where ω(k) =−k 3 ,
by Fourier transforms is as follows:
u(x, 0)
Fourier Transform
✲
û(k, 0)
u(x, t)
✛
Inverse Fourier Transform
❄
û(k, t) =û(k, 0)e −iω(k)t
In fact, as we shall discuss in Sect. 3, the method for solving nonlinear wave
equations, such as KdV and KP, which is referred to as the Inverse Scattering
Transform (IST), is in many ways a natural generalization of Fourier transforms.
2 IST for Nonlinear Equations in 1+1 Dimensions
The KdV equation (1.15) was the first equation solved (on the infinite line
with appropriately decaying data) by inverse scattering methods [5]. Subse-