The mKdV equation on a finite interval
Abstract
We analyse an initial-boundary value problem for the mKdV equation on a finite interval by expressing the solution in terms of the solution of an associated matrix Riemann-Hilbert problem in the complex
k
-plane. This Riemann-Hilbert problem has explicit
(x,t)
-dependence and it involves certain functions of
k
referred to as ``spectral functions''. Some of these functions are defined in terms of the initial condition
q(x,0)=
q
0
(x)
, while the remaining spectral functions are defined in terms of two sets of boundary values. We show that the spectral functions satisfy an algebraic ``global relation'' that characterize the boundary values in spectral terms.