Classification and stability of infinite soliton trains and kink-soliton trains of nonlinear Schrödinger equations, and the construction of multikink and terrace solutions

The survey of Le Coz and Tsai (arXiv:1409.8379, 2014/2016), reviewing their joint constructions with Li, established existence — for large relative speeds, and uniqueness only in a strong-decay class — of infinite soliton trains and kink-soliton trains for non-integrable nonlinear Schrödinger equations, and posed the following open problems: classification of multi-solitons beyond the strong-decay class (including profiles with linearly unstable constituents, kink-soliton trains, and infinite trains), stability of infinite trains and kink-soliton trains, and construction of multikinks connecting nonzero constants on both sides and of terrace-shaped solutions; among many other related open problems, the survey also remarks on constructing solutions when one of the composing elements is a line-soliton in two dimensions. Later work resolved parts of the finite-configuration picture — conditional uniqueness and spatial decay of multi-solitons (Côte–Friederich), energy-class uniqueness in the mass-critical case (Cao–Su–Zhang), orbital and asymptotic stability of finite dark-soliton chains for general nonintegrable one-dimensional NLS with nonzero far field (Berthoumieu), asymptotic stability of multi-solitons for one-dimensional supercritical NLS on center-stable manifolds (Chen–Moutinho) — and Lin and Tsai constructed mixed-dimensional infinite trains containing line solitons in two dimensions. However, no result addresses the classification or stability of the infinite trains and kink-soliton trains themselves, nor the construction of multikink or terrace solutions; the problem remains open.

Background

The nonlinear Schrödinger equation (NLS)

$iu_t+\Delta u+f(u)=0\quad\text{on }\mathbb R\times\mathbb R^d$

with a focusing, energy-subcritical nonlinearity such as $f(u)=|u|^{p-1}u$ is a canonical nonlinear dispersive equation, well posed in the energy space $H^1(\mathbb R^d)$. By the soliton resolution conjecture — the guiding open heuristic of the field — a generic global solution is expected to decompose at large time into a dispersive (scattering) part plus well-separated non-scattering structures. The elementary non-scattering structures are: solitons, that is, localized bound states $\phi$ of the stationary elliptic equation $-\Delta\phi+\omega\phi-f(\phi)=0$, boosted by Galilean invariance to travel at constant velocity; dark solitons, which arise for defocusing nonlinearities with a nonzero background, such as the Gross–Pitaevskii equation $f(u)=(1-|u|^2)u$, and are travelling waves whose modulus tends to a nonzero constant at infinity; and kinks, one-dimensional travelling waves whose profile has different limits at $-\infty$ and $+\infty$ (for the Gross–Pitaevskii equation the explicit family $\phi_K(x)=\sqrt{(2-c^2)/2}\,\tanh(x\sqrt{2-c^2}/2)+ic/\sqrt{2}$, $|c|<\sqrt{2}$; the member $c=0$ is the black soliton connecting $-1$ to $1$).

A multi-soliton is a solution that converges as $t\to+\infty$, in a space-time norm, to a finite sum $R=\sum_{j=1}^N R_j$ of solitons with distinct velocities; an infinite soliton train is the analogous object with countably many solitons whose parameters satisfy summability conditions; a kink-soliton train is a one-dimensional solution with one or two kinks attached to a finite or infinite soliton train. Finite multi-solitons of NLS were first constructed by energy methods (Martel and Merle, 2006), and at high speed even with excited-state constituents (Côte and Le Coz, 2011) — excited states meaning bound states of the stationary elliptic equation other than ground states, which are linearly unstable — who also showed that when a constituent soliton is linearly unstable a one-parameter family of multi-solitons converges to the same prescribed profile, so uniqueness cannot be expected in general. Because infinite trains and kink-soliton trains carry infinite energy, energy methods fail for them; Le Coz, Li and Tsai (2015) and Le Coz and Tsai (2014) instead constructed, by a fixed-point argument around the desired profile combined with Strichartz estimates (standard space-time dispersive estimates for the linear Schrödinger flow), under assumptions of sufficiently large relative speeds: finite and infinite soliton trains in any dimension, finite kink-soliton trains, and infinite kink-soliton trains in one dimension. Uniqueness, however, was proved only in the class of solutions decaying strongly (exponentially in time) towards the prescribed profile, so solutions with weaker convergence to the same profile are not excluded.

Regarding dynamics, a configuration is orbitally stable if data close to it remain, for all later time, close to some modulation (translation, phase shift) of it, and asymptotically stable if the solution in fact converges to a modulated configuration plus dispersion. At the time of the survey, stability was known only for finite configurations: $H^1$ orbital stability of sums of $K$ solitons under hypotheses on the nonlinearity (high regularity or flatness at the origin) that exclude the power-type nonlinearities (Martel, Merle and Tsai, 2006); asymptotic stability under similarly restrictive hypotheses (Perelman, 2004; Rodnianski, Schlag and Soffer, 2003); and, among power-type nonlinearities, only the chains of dark solitons of the one-dimensional Gross–Pitaevskii equation in the energy space (Béthuel, Gravejat and Smets, 2014). On the classification side, the complete programme — multi-solitons converging to a fixed $N$-soliton profile form an $N$-parameter family, one free parameter per unstable direction — had been carried out only for the generalized Korteweg–de Vries equation (Combet, 2011), with partial multi-existence results for NLS (Combet, 2014). The survey of Le Coz and Tsai concludes by listing what remains open for the newly constructed objects, and this record concerns that list.

Problem Statement

For nonlinear Schrödinger equations $iu_t+\Delta u+f(u)=0$ with the general (non-integrable) nonlinearities considered in the source survey — for which Le Coz, Li and Tsai and Le Coz and Tsai constructed, at large relative speeds, finite and infinite soliton trains and kink-soliton trains that are unique only in the class of solutions decaying strongly towards the prescribed profile — settle the following linked open problems, which together form the completion of the existence theory of these objects into a classification and stability theory:

(1) Classification. For a prescribed profile — a finite sum of solitons (including sums containing linearly unstable or excited-state solitons), a kink-soliton train, or an infinite soliton train with parameters satisfying the construction assumptions — classify all global solutions converging to that profile as $t\to+\infty$ in a natural class (such as the energy space or a prescribed algebraic decay class), beyond the strong-decay class in which uniqueness already holds. In particular, decide whether, as for the generalized Korteweg–de Vries equation, the solutions converging to a fixed $N$-soliton sum form an $N$-parameter family when unstable solitons are present, and settle uniqueness versus multi-existence for kink-soliton train and infinite train profiles, for which no result is available.

(2) Stability. Decide the orbital or asymptotic stability, or instability, of the infinite soliton trains and the kink-soliton trains themselves — the source states this is completely open, the known stability results (before and since) concerning only finite multi-solitons under restrictive hypotheses on the nonlinearity and finite chains of dark solitons.

(3) Construction of the missing configurations. Construct multikink solutions connecting nonzero constants on both sides and solutions whose modulus has a terrace shape (a monotone sequence of increasing plateaus), for which no appropriate ansatz is currently known.

The items are linked by one scientific objective — determining which multi-soliton-type asymptotic states of non-integrable NLS exist, are uniquely determined by their profile, and are dynamically stable — and each item is a component of that objective as posed by the source.

References

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