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Analysis of the global relation for the nonlinear Schrödinger equation on the half-line. (English) Zbl 1055.35107
Summary: It has been shown recently that the unique, global solution of the Dirichlet problem of the nonlinear Schrödinger equation on the half-line can be expressed through the solution of a \(2\times 2\) matrix Riemann-Hilbert problem. This problem is specified by the spectral functions \(\{a(k),b(k)\}\) which are defined in terms of the initial condition \(q(x,0)=q_0(x)\), and by the spectral functions \(\{A(k),B(k)\}\) which are defined in terms of the specified boundary condition \(q(0,t)=g_0(t)\) and the unknown boundary value \(q_x(0,t)=g_1(t)\). Furthermore, it has been shown that given \(q_0\) and \(g_0\), the function \(g_1\) can be characterized through the solution of a certain ’global relation’ coupling \(q_0\), \(g_0\), \(g_1\), and \(\Phi(t,k)\), where \(\Phi\) satisfies the \(t\)-part of the associated Lax pair evaluated at \(x=0\).
We show here that, by using a Gelfand-Levitan-Marchenko triangular representation of \(\Phi\), the global relation can be explicitly solved for \(g_1\).

MSC:
35Q55 NLS equations (nonlinear Schrödinger equations)
37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
35Q15 Riemann-Hilbert problems in context of PDEs
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