Integrability of Nonlinear Systems

Nonlinear Waves, Solitons, and IST 17
There are a number of results which follow from the above developments.
a) For scattering data that correspond to potentials satisfying (3.3), the solution
to the GLM equation exists. With suitable conditions on u and its derivatives,
global solutions to the KdV equation can be established. b) Long-time asymptotic
analysis of the KdV equation can be ascertained. The solution is comprised
of a discrete part consisting of N soliton (see below) waves moving to the right,
and a dispersive tail which decays algebraically as t →∞. c) The discrete part
of the spectrum can be solved in terms of a linear algebraic system. In the GLM
equation, the discrete spectrum corresponds to a degenerate kernel. From the
RHBVP, the following linear system results,
N l (x, t)+
N∑
p=1
C p (0)
i(κ p + κ l ) exp(−2κ px +8κ 3 pt)N p (x, t) = exp(−2κ l x) (3.36)
where N l (x, t) ≡ N(x, k = iκ l ,t), and from the solution of (3.36) we reconstruct
the solution of KdV, u(x, t), via
u(x, t) =2i ∂
∂x
N∑
C p (t)N p (x, t). (3.37)
p=1
A one-soliton solution (N = 1) is given by
u(x, t) =2κ 2 1 sech 2 κ 1 (x − 4κ 2 1t − x 1 ) (3.38)
where C 1 (0)=2κ 1 exp(2κ 1 x 1 ), and a two-soliton solution (N = 2) is given by
u(x, t) = 4(κ2 2 − κ 2 1)[(κ 2 2 − κ 2 1)+κ 2 1 cosh(2κ 2 ξ 2 )+κ 2 2 cosh(2κ 1 ξ 1 )]
[(κ 2 − κ 1 ) cosh(κ 1 ξ 1 + κ 2 ξ 2 )+(κ 2 + κ 1 ) cosh(κ 2 ξ 2 − κ 1 ξ 1 )] 2 , (3.39)
where ξ i = x − 4κ 2 i t − x i,C i (0) = 2κ i exp(2κ i x i ), i = 1, 2. The two-soliton
solution shows that the sum of two solitary waves of the form given by (3.38) is
the asymptotic state of (3.39), but there is a phase shift due to the interaction.
We also note that knowledge that the function a(k, t) is a constant of the
motion can be related to the infinite number of conservation laws of KdV (cf. [3,
7]).
It should also be noted that discretizations of (2.5), (3.1) lead to interesting
discrete nonlinear evolution equations which can be solved by IST. The best
known of these equations is the Toda lattice,
∂ 2 u n
∂t 2 = exp(−(u n − u n−1 )) − exp(−(u n+1 − u n )), (3.40)
which is related to the linear discrete Schrödinger scattering problem,
α n v n+1 + α n−1 v n−1 + β n v n = kv n , (3.41)