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Robin boundary condition and shock problem for the focusing nonlinear Schrödinger equation. (English) Zbl 1420.35364

Summary: We consider the initial boundary value (IBV) problem for the focusing nonlinear Schrödinger equation in the quarter plane \(x>0\), \(t>0\) in the case of periodic initial data, \(u(x,0) = \alpha\operatorname{exp}(-2i\beta x)\) (or asymptotically periodic, \(u(x, 0) =\alpha \operatorname{exp}(-2i\beta x)\rightarrow 0\) as \(x\rightarrow\infty\)), and a Robin boundary condition at \(x = 0: u_x(0, t)+qu(0,t) = 0\), \(q\neq0\). Our approach is based on the unified transform (the Fokas method) combined with symmetry considerations for the corresponding Riemann-Hilbert (RH) problems. We present the representation of the solution of the IBV problem in terms of the solution of an associated RH problem. This representation also allows us to determine an initial value (IV) problem, of a shock type, a solution of which being restricted to the half-line \(x>0\) is the solution of the original IBV problem. In the case \(\beta < 0\), the large-time asymptotics of the solution of the IBV problem is presented in the “rarefaction” sector, demonstrating, in particular, an oscillatory behavior of the boundary values in the case \(q>0\), contrary to the decay to 0 in the case \(q<0\).

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35Q15 Riemann-Hilbert problems in context of PDEs
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
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References:

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