Asymptotic stability of N-solitons in the cubic NLS equation

Saalmann, Aaron

doi:10.1142/s0219891617500151

Cited by 4 publications

(4 citation statements)

References 15 publications

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“…The inverse scattering transform and the reconstruction formulas for the global solutions (u, v) to the MTM system (1.1) can be used to solve other interesting analytical problems such as longrange scattering to zero [6], orbital and asymptotic stability of the Dirac solitons [9,28], and an analytical proof of the soliton resolution conjecture. Similar questions have been recently addressed in the context of the cubic NLS equation [8,10,30] and the derivative NLS equation [17,23].…”

Section: Introductionmentioning

confidence: 65%

Inverse Scattering for the Massive Thirring Model

Pelinovsky

Saalmann

2019

Nonlinear Dispersive Partial Differential Equations and Inverse Scattering

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We consider the massive Thirring model in the laboratory coordinates and explain how the inverse scattering transform can be developed with the Riemann-Hilbert approach. The key ingredient of our technique is to transform the corresponding spectral problem to two equivalent forms: one is suitable for the spectral parameter at the origin and the other one is suitable for the spectral parameter at infinity. Global solutions to the massive Thirring model are recovered from the reconstruction formulae at the origin and at infinity.A.S. gratefully acknowledges financial support from the project SFB-TRR 191 "Symplectic Structures in Geometry, Algebra and Dynamics" (Cologne University, Germany).

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Section: Introductionmentioning

confidence: 65%

Inverse Scattering for the Massive Thirring Model

Pelinovsky

Saalmann

2019

Nonlinear Dispersive Partial Differential Equations and Inverse Scattering

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“…which is a consequence of our assumtion (8.12). Then, by the arguments of Lemmas 6.1 and 6.3 in [Saa17a] we obtain the following result: the meromorphic function…”

Section: Soliton Resolutionmentioning

confidence: 79%

“…The proof of Theorem 8.1 can be obtained from the proofs of Lemma 3.1 and 3.2 in [Saa17a] applied to RHP's 2.5 and 2.6 separately. The technique of steepest descent is not used for this result, because the asymptotic functions u K and v K in (8.6) still have a non-vanishing reflection coefficient p given by (8.5) or, equivalently, (8.7).…”

Section: Analysis Of Pure ∂ Problemmentioning

confidence: 99%

Long-time asymptotics for the Massive Thirring model

Saalmann

2018

Preprint

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We consider the massive Thirring model and establish pointwise long-time behavior of its solutions in weighted Sobolev spaces. For soliton-free initial data we can show that the solution converges to a linear solution modulo a phase correction caused by the cubic nonlinearity. For initial data that support finitely many solitons we obtain long-time behavior in the form of a multi-soliton which in turn splits into a sum of localized solitons. This phenomenon is known as soliton resolution. The methods we will is the nonlinear steepest descent of Deift and Zhou and the paper also relies on recent progress in the inverse scattering transform for the massive Thirring model.

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“…Remark 1.5. In dimension 1, well known is the case when β(|u| 2 )u = −|u| 2 u, where it is possible to apply methods from the theory of integrable systems [12,117], which require u 0 s.t. x s u 0 ∈ L 2 (R) for s > 1/2, see [41].…”

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confidence: 99%