initial boundary value problems for integrable systems (IBVP)

The principal question is to determine the long-time behavior of a solution of the focusing nonlinear Schrödinger equation on the half-line, with prescribed initial profile $f$ and oscillatory Dirichlet data of frequency $w\in\mathbb R$:

$$i u_t=u_{xx}+2|u|^2u,\qquad x\geq 0,\quad t\geq 0,$$ $$u(x,0)=f(x),\qquad x\geq 0;\qquad u(0,t)=e^{iwt},\qquad t\geq 0.$$

The question is what behavior $u(x,t)$ exhibits as $t\to\infty$ under this boundary forcing. Deift, Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems II (2017)

Background

Fokas introduced the unified transform in the late 1990s as a flexible inverse-scattering method for boundary-value problems. In its formulation of half-line NLS, the usual initial and Dirichlet data are generally accompanied by the Neumann trace $u_x(0,t)$. The source emphasizes that this makes the general problem overdetermined from the method’s perspective: special integrable boundary conditions can make the missing trace accessible, but arbitrary Dirichlet forcing need not do so. Although the transform yields a Riemann–Hilbert formulation, the nonlinear steepest-descent analysis used for long-time asymptotics requires the Neumann data to be known a priori in the situation described there. Deift, Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems II (2017)

The motivating picture is that periodic forcing at one endpoint can excite waves that travel into an extended system. Deift compares this with shaking a rope and discusses a driven Toda lattice as a phenomenological analogue. For focusing NLS, the proposed analogy suggests that periodic, finite-gap NLS waves might emerge as the driving frequency decreases. This is presented as an expectation and an open direction, not as an established asymptotic description of the stated NLS problem. Deift, Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems II (2017)

Known results

For the oscillatory Dirichlet problem itself, the source reports partial results by Fokas and Lenells in the limit $w\to\infty$. It does not give a general long-time description for prescribed $w$ and initial profile $f$. Thus the reported large-frequency results should not be read as resolving the question for arbitrary frequency or as establishing the anticipated emergence of finite-gap waves. Deift, Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems II (2017) Fokas and Lenells (2012)

Robin boundary conditions provide a contrasting integrable case. For focusing NLS, $u_x(0,t)+q u(0,t)=0$ with fixed $q$, the unified-transform approach gives an effective Riemann–Hilbert representation. Deift and Park instead used the earlier Bikbaev–Tarasov method: a Bäcklund transformation extends the half-line solution across the boundary, converting the problem to one on the whole line. Their work treats long-time asymptotics in the setting connected to a delta potential and even initial data. These are Robin-boundary results, not a solution of the oscillatory Dirichlet problem. Deift, Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems II (2017) Bikbaev and Tarasov (1991) Deift and Park (2011) Its and Shepelsky (2013)

The same discussion reports that, for focusing NLS with smooth initial data and nonzero Robin parameter, Deift and Park establish classical regularity using the Bäcklund method; the case $q=0$ follows by reflection. It identifies control of singularities in the corresponding Bäcklund extension for defocusing NLS with Robin data as an open problem. More broadly, it poses classical smoothness for Robin problems with general smooth nonlinearities, and for dispersive initial-boundary value problems, as PDE questions rather than asserting general regularity results. Deift, Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems II (2017) Deift and Park (2011)