The unified method. II: NLS on the half-line with \(t\)-periodic boundary conditions (Q2890762)

For Part I see Zbl 1256.35044; for Part III see Zbl 1256.35046.NEWLINENEWLINEIn this second paper, the authors implement the explained earlier Riemann-Hilbert (RH) technique to the initial boundary value problem (IBVP) for the NLS equation in \((x,t)\in{\mathbb R}_+\times{\mathbb R}_+\). The main difficulty in the discussed approach is the determination of the jump matrices for the RH problem in terms of the boundary data in a well-posed IBVP. Indeed, the entries of the jump matrices can be found solving a system of linear integral equations determined by the values \(q(0,t)\) and \(q_x(0,t)\) of the solution \(q(x,t)\) to the NLS equation. In contrast, in the well posed IBVP with the Dirichlet boundary condition, the function \(q_x(0,t)\) remains unknown.NEWLINENEWLINEHere, the authors express \(q_x(0,t)\) in terms of \(q(0,t)\) and two auxiliary functions satisfying a system of nonlinear integral equations and appearing in the Gelfand-Levitan-Marchenko representation of the NLS \(\Psi\)-function. This form of the generalized Dirichlet to Neumann map significantly simplifies the previously known one. This form also supports a perturbative scheme for computing \(q_x(0,t)\) in the case of the ``small'' Dirichlet data and allows the authors to apply this scheme to the large \(t\) asymptotic study of the IBVP with the asymptotically periodic Dirichlet boundary condition. Namely, in the case of the trivial initial condition, \(q(x,0)\equiv0\), and the Dirichlet boundary data \(q(0,t)=\epsilon g_0(t)+{\mathcal O}(\epsilon^2)\), where \(\dot g_0(t+t_p)-\dot g_0(t)={\mathcal O}(t^{-\frac{1}{2}-\nu})\), \(\nu>0\), \(t\to\infty\), the boundary value \(q_x(0,t)\) becomes asymptotically \(t_p\)-periodic in the 1st order of the perturbative expansion. Furthermore, if \(q(0,t)=\epsilon\sin(t)+{\mathcal O}(\epsilon^4)\), then \(q_x(0,t)\) becomes asymptotically \(2\pi\)-periodic up to the 3rd order of the perturbative expansion.