A Riemann-Hilbert Approach to the Chen-Lee-Liu Equation on the Half Line

In this paper, the Fokas unified method is used to analyze the initial-boundary value for the Chen- Lee-Liu equation on the half line (-a, 0] with decaying initial value. Assuming that the solution u(x, t) exists, we show that it can be represented in terms of the solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter lambda. The jump matrix has explicit (x, t) dependence and is given in terms of the spectral functions {a(lambda), b(lambda)} and {A(lambda), B(lambda)}, which are obtained from the initial data u (0)(x) = u(x, 0) and the boundary data g (0)(t) = u(0, t), g (1)(t) = u (x) (0, t), respectively. The spectral functions are not independent, but satisfy a so-called global relation.