Long-time behavior with non-generic initial data

For focusing nonlinear Schrödinger (NLS), the question concerns whether the non-generic spectral behavior described below changes the long-time behavior of solutions, and, in the communication-cable model, whether those changes are measurable. In the setup, choose any $z_0>0$ and an arbitrarily small open disk $D$ centered at $z_0$ with $D\subset\{z\in\mathbb C:\operatorname{Re}z>0\}$. Write $D^+=D\cap\{z\in\mathbb C:\operatorname{Im}z>0\}$, take $u$ analytic in $D^+$ and continuous on its closure, and set

$$B=\{z\in D^+:u(z)=0\}.$$

The stated construction is an AKNS operator $T$ with infinitely smooth, rapidly decaying coefficients for which every point of $B$ is an $L^2(\mathbb R)$ eigenvalue. The research question is: what effect do such eigenvalues, which can accumulate on the real axis at essentially arbitrary rates, have on the long-time behavior of the associated NLS solution, and are any effects measurable? Deift, Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems (2007)

Background

Long-time asymptotic analysis by the Riemann–Hilbert and steepest-descent methods depends on the spectral data of the Lax operator. The 2007 discussion contrasts focusing NLS with KdV, modified KdV, defocusing NLS, and the Toda lattice, for which it describes detailed asymptotics for general initial data. For generic focusing-NLS data, the spectrum is described as the real line with absolutely continuous spectrum, together with finitely many simple eigenvalues off the real axis; these discrete eigenvalues correspond to solitons. The issue raised here is that smooth, rapidly decaying coefficients do not by themselves rule out a much less regular discrete spectrum. The same discussion notes a related challenge for the Boussinesq equation: its Lax operator is self-adjoint but third order, and even its generic-data long-time behavior is described as only partially understood. Deift, Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems (2007)

Known results

A later focusing-NLS preprint treats one important non-generic case: high-order discrete spectrum. For initial data $q_0\in H^{1,1}(\mathbb R)$ whose scattering data have double zeros of the scattering coefficient, its theorem gives an asymptotic expansion on any fixed space-time cone as $|t|\to\infty$. In the notation of that result, the solution is a cone-dependent soliton term plus a $t^{-1/2}$ correction, with error $O(t^{-3/4})$. This establishes long-time asymptotics for that specified high-order-pole setting; it does not address the question’s construction with eigenvalues accumulating on the real axis at an arbitrary rate. Long time asymptotics for the focusing nonlinear Schrödinger equation in the solitonic region with the presence of high-order discrete spect

A distinct, more recent result studies a large-number limit rather than long-time evolution from one fixed datum: for focusing-NLS $N$-soliton solutions whose eigenvalues accumulate on two bounded horizontal segments and whose norming constants stay bounded away from zero, the solution is asymptotically described by a rapidly oscillatory elliptic wave on compact subsets of space-time as $N\to\infty$. This shows that eigenvalue accumulation can have a structured asymptotic manifestation in that soliton-gas regime, but its hypotheses and limit differ from the real-axis accumulation and long-time question posed here. Gkogkou, Mazzuca, and McLaughlin (2025)