Initial/boundary value problems for integrable systems

Given initial data $u_0(x)$, $x\geq 0$, smooth and compactly supported, and boundary data $u(0,t)=\sin(\omega t)$, $t\geq 0$, with $\omega\neq 0$, the question is: suppose $u(x,t)$ solves NLS in $(x\geq 0,t\geq 0)$ with these data. How does $u(x,t)$ behave as $t\to\infty$, for arbitrary such $u_0$? This is the long-time behavior problem as posed; the NLS equation and the intended precise form of the asymptotic description are not further specified. Deift, Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems (2007)

Background

Fokas introduced the unified approach to inverse scattering in the late 1990s. Applied to half-line initial-boundary value problems, it brings both the initial profile and boundary traces into the spectral formulation. For a Dirichlet problem, the prescribed trace is $g_0(t)=u(0,t)$, while the Neumann trace $g_1(t)=u_x(0,t)$ is dependent data that also enters the solution construction. Recovering that trace from the initial and prescribed boundary data is therefore a central step toward using the method for long-time asymptotics. Deift, Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems (2007) The Unified Method: I Non-Linearizable Problems on the Half-Line

The question was posed as an obstacle to long-time analysis, not as a claim that the boundary trace cannot be determined. Subsequent unified-method work gives an intrinsic finite-time characterization of the unknown trace. This distinction matters: recovering boundary data on a finite time interval does not, by itself, provide the estimates on that trace needed to derive asymptotics as time tends to infinity. Deift, Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems (2007) The Unified Method: I Non-Linearizable Problems on the Half-Line

Known results

For the NLS Dirichlet problem on a finite interval $0<t<T$, Theorem 3.2 of The Unified Method: I Non-Linearizable Problems on the Half-Line characterizes the spectral functions using a system of nonlinear integral equations for auxiliary functions $\Phi_1$ and $\Phi_2$. Under its hypotheses—Schwartz-class initial profile, sufficiently smooth boundary data compatible with the initial data at the corner, finitely many simple zeros of $a(k)$ (possibly none), and no zeros on the boundaries of the specified spectral domains—it also gives an integral representation for the unknown Neumann value $g_1(t)$. The paper describes the resulting nonlinear system as an effective characterization; its discussion conditions the determination on unique solvability of that system. This is a finite-time result, not a long-time asymptotic formula for the sinusoidal-boundary problem. The Unified Method: I Non-Linearizable Problems on the Half-Line

For periodic boundary behavior, later work studies the Dirichlet-to-Neumann relation at large times as a prerequisite for rigorous long-time asymptotics. In particular, it considers quarter-plane solutions whose Dirichlet and Neumann traces each approach smooth periodic functions with errors $O(t^{-7/2})$. This provides a framework for asymptotically periodic data, but does not establish the requested asymptotics for the original problem with arbitrary smooth compactly supported $u_0$ and prescribed sinusoidal Dirichlet data. The nonlinear Schrödinger equation with t-periodic data: I. Exact results