Stability in the energy space for chains of solitons of the one-dimensional Gross-Pitaevskii equation

Abstract

We establish the stability in the energy space for sums of solitons of the one-dimensional Gross-Pitaevskii equation when their speeds are mutually distinct and distinct from zero, and when the solitons are initially well-separated and spatially ordered according to their speeds.

Stability in the energy space for chains of solitons of the Landau–Lifshitz equation

“…The proof is reminiscent of the one in [3,Proposition 1]. For the sake of completeness, we provide the following details.…”

“…Our strategy is reminiscent of the one developed to tackle the stability of well-prepared chains of solitons for the generalized Korteweg-de Vries equations [24], the nonlinear Schrödinger equations [25], or the Gross-Pitaevskii equation [3].…”

“…Since the operator norm of the inverse matrices M(·, t) −1 is bounded by some positive number depending only on c * , we can apply a density argument to derive from (3.1) the C 1 nature of the modulation functions t → a(t) and t → c(t), as well as estimate (48). We refer to [3] for more details. This concludes the proof of Proposition 4.…”

“…Originally introduced by Landau and Lifshitz in [21], this equation describes the dynamics of magnetization in a one-dimensional ferromagnet, typically in samples of CsNiF 3 and TMNC (see e.g. [19,26,17]).…”

Asymptotic stability of the black soliton for the Gross–Pitaevskii equation

“…In other words, there is no common speed for solitons and dispersion. In contrast, for |c| < √ 2, the non-constant solutions are uniquely given by the formula 4) up to the invariances of the problem, that is, multiplication by a constant of modulus one and translation. Solitons U c with speed c = 0 do not vanish on R. They are called dark solitons, with reference to non-linear optics where |Ψ| 2 refers to the intensity of light.…”