The Fokas-Lenells equation on the finite interval

The Fokas unied method for solving initial boundary value problem for nonlinear integrable PDEs was announced in [1], and further developed in [2] and [3]. The main work of this method is construction of solution in terms of matrix Riemann Hilbert (RH)problem, which is based on the spectral analysis of the two linear eigenvalue equations so called Lax pairs, and analysis of global relation, which is constructed by spectral functions.There exits much difficulty for the latter that there are only some boundary value conditions prescribed. For a particular class of boundary conditions, called linearizable. There are many literature such as [4] and [5], have studied this linearizable boundary conditions for equations on the half line, for which it is possible to bypass the difficulty of computing the unknown boundary values. The other one is nonlinearizable boundary conditions that some unknown boundary value data can be expressed by known boundary value conditions such as significant developments [6], [7] and[8]