Integrability of Nonlinear Systems

Nonlinear Waves, Solitons, and IST 13
Indeed, (3.11) is a fundamental equation in this approach. It is a generalized
Riemann-Hilbert boundary value problem (RHBVP), which is a consequence of
the following facts:
i) M(x, k) and a(k) can be analytically extended to the upper half k-plane
(UHP) and tend to unity as |k| →∞(Im k>0).
ii) ¯N(x, k) can be analytically extended to the lower half k-plane (LHP) and
tends to unity as |k| →∞ (Im k0,
θ(x) =
0ifx0). In fact, the zeroes of a(k) are relevant in what follows. It can be
established that for u(x) real there are a finite number of simple zeroes, all of