Perturbation theory for infinite dimensional integrable systems

The research direction is to understand the long-time behavior of perturbations of focusing cubic NLS near a multisoliton. Replacing the defocusing cubic term in the model equation by its focusing counterpart gives

$$i u_t+u_{xx}+2|u|^2u-\varepsilon V(|u|)u=0.$$

The proposed regime has small positive $\varepsilon$ and a perturbation satisfying $V(r)\sim r^p$ as $r\to0$, for $p>2$ sufficiently large. The question posed is to describe solutions when the initial data lie near a $k$-soliton solution of the unperturbed focusing cubic NLS, in particular for $k\ge2$, including their long-time asymptotics. This is a research direction rather than a theorem with a specified neighborhood or a fully quantified class of admissible $V$. Deift, Some open problems in random matrix theory and the theory of integrable systems (Problem 12)

Background

The scattering transform provides the basic coordinate system for the line problem. For NLS, the scattering data evolve by a simple oscillatory phase, while the inverse transform can be formulated as a Riemann–Hilbert problem. Zhou’s mapping theorem gives a bi-Lipschitz correspondence between the weighted Sobolev spaces $H^{k,j}$ and $H^{j,k}$ for $k\ge0$ and $j\ge1$. Deift and Zhou used this framework, together with nonlinear steepest descent for the associated Riemann–Hilbert problems, to control long-time behavior without losing derivatives. In the defocusing setting, decay in the uniform norm makes a perturbation of order higher than cubic increasingly small relative to the cubic term. Deift and Zhou (2002)

The focusing problem differs in a crucial way: solitons do not decay uniformly in time, so the perturbation need not become negligible relative to the cubic nonlinearity along a soliton. This is why the multisoliton question is not a direct extension of the defocusing analysis. The line and periodic settings also have different mechanisms: dispersion drives decay on the line, whereas the periodic problem lacks room for such spatial dispersal. In the periodic setting, KAM methods instead yield persistence of certain finite-dimensional invariant tori under Hamiltonian perturbations. Deift, Some open problems in random matrix theory and the theory of integrable systems (Problem 12) Deift and Zhou (2002)

Known results

For defocusing NLS, Deift and Zhou proved long-time results for a class of perturbed equations on the line. In their weighted $H^{1,1}$ framework, the wave operators exist on nonempty open sets and conjugate the perturbed flow to the unperturbed NLS flow. Their analysis yields NLS-type long-time asymptotics and, on the relevant open invariant sets, complete integrability in the sense of a complete family of commuting integrals. These conclusions are conditional on the stated phase-space and perturbation hypotheses; they do not establish the corresponding focusing result near solitons. Deift and Zhou (2002)

A separate advance concerns stability for the unperturbed focusing equation. Koch and Tataru proved that the pure $N$-soliton states form a uniformly smooth manifold and are uniformly stable in $H^s$ for every $s>-\tfrac12$. The stability estimate holds for NLS and the commuting mKdV flow, uniformly when the spectral parameters range over a compact subset of the open upper half-plane, and includes multisolitons with repeated spectral parameters. This supplies a strong stability result for the unperturbed multisoliton family, but it does not analyze the perturbed equation or provide its long-time asymptotics. Koch and Tataru (2024)