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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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