The unified method. III: Nonlinearizable problems on the interval (Q2890763)

For Part I see Zbl 1256.35044; for Part II see Zbl 1256.35045.NEWLINENEWLINEThe third and the last paper of the series is devoted to the Riemann-Hilbert (RH) approach to the initial-boundary value problem (IBVP) for the NLS equation in \((x,t)\in[0,L]\times[0,T]\). The main difference between the present case and those considered in two previous papers is the increased number of the so-called spectral functions determining the jump matrices in the relevant RH problem. The authors construct the Dirichlet to Neumann and Neumann to Dirichlet maps expressing the boundary values of the normal derivatives \(q_x(0,t)\) and \(q_x(L,t)\) in terms of \(q(0,t)\) and \(q(L,t)\) and vice versa. These maps are used to construct the jump matrices in the RH problem for the well-posed (Dirichlet or Neumann) IBVPs and allow the authors to solve these IBVPs in terms of four auxiliary functions satisfying the system of nonlinear integral equations. The authors show that their formulas reproduce those in the first paper of the series obtained for \(L=\infty\). They also show that these formulas are effective in the sense they support a perturbative scheme for the case of the ``small'' boundary values.