The unified method. I: Nonlinearizable problems on the half-line (Q2890761)

For Part II see Zbl 1256.35045; for Part III see Zbl 1256.35046.NEWLINENEWLINEThis is the first in the series of papers devoted to the solution of the initial boundary value problems (IBVPs) on the half-line for linear and integrable nonlinear evolution PDEs using an extension of the inverse scattering transform technique. This method utilizes the Zakharov-Shabat pair for the PDE, \(d\Psi=\Omega\Psi\), and the fact that the domain for the solution of the IBVP is a infinite convex polygon. In more detail, if the diagonal part of the \(1\)-form \(\Omega\) is exact, \(\Omega_{diag}=df\), then the gauge transform \(\mu=e^{-f}\Psi\) satisfies the equation \(d\mu=e^{-f}(\Omega-\Omega_{diag})e^{f}\mu\). The integrals of the latter equation, \(\mu_j(x,t)=I+\int_{(x_j,t_j)}^{(x,t)}e^{-f}(\Omega-\Omega_{diag}) e^{f}\mu_j\), along the paths initiated at the vertices \((x_j,t_j)=(0,0),(0,T),(\infty,t)\) of the infinite convex polygon in the real \((x,t)\)-plane yield the collection of functions \(\mu_j(x,t)\) that is used to formulate a boundary Riemann-Hilbert (RH) problem in the complex plane of the spectral variable \(k\). The latter RH problem allows one to compute the solution of the IBVP.NEWLINENEWLINEThe main difficulty in this approach is the description of the so-called spectral functions determining the jump matrices in the above RH problem. Some of the necessary data, the spectral functions \(a(k),b(k)\), are determined by the initial data in the IBVP solving a 1-dimensional linear ODE. Other data, the spectral functions \(A(k),B(k)\), are determined by the boundary values in IBVP solving another linear ODE subject to additional analytic constraints. For certain kind of boundary conditions called ``linearizable'', all the spectral functions can be found explicitly. For the ``nonlinearizable'' boundary conditions, the spectral functions satisfy a system of nonlinear integral equations. Such a characterization of the spectral functions is called ``effective'' if in the linear limit it yields a solution of the linearizable IBVP and if it supports a perturbative approach to the solution of the nonlinearizable IBVP.NEWLINENEWLINEIn the present paper, the authors solve the Dirichlet and Neumann boundary value problems for the NLS and mKdV equations on the spatial half-line for the finite times finding the effective characterizations of the corresponding spectral functions.