Global well-posedness for the derivative non-linear Schrödinger equation

Jenkins, Robert M.; Liu, Jiaqi; Perry, Peter; Sulem, Catherine

doi:10.1080/03605302.2018.1475489

Cited by 35 publications

(30 citation statements)

References 21 publications

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“…There is also a number of works [15,16,17,29,30] where the global well-posedness of the DNLS equation was studied by means of the inverse scattering techniques. The corresponding results get rid of the smallness assumption on the mass, but require more regularity and decay on the initial data.…”

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confidence: 99%

Global well-posedness for the derivative nonlinear Schrödinger equation

Bahouri¹,

Perelman²

2020

Preprint

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This paper is dedicated to the study of the derivative nonlinear Schrödinger equation on the real line. The local well-posedness of this equation in the Sobolev spaces H s (R) is well understood since a couple of decades, while the global well-posedness is not completely settled. For the latter issue, the best known results up-to-date concern either Cauchy data in H 1 2 (R) with mass strictly less than 4π or general initial conditions in the weighted Sobolev space H 2,2 (R). In this article, we prove that the derivative nonlinear Schrödinger equation is globally well-posed for general Cauchy data in H 1 2 (R) and that furthermore the H 1 2 norm of the solutions remains globally bounded in time. One should recall that for H s (R), with s < 1/2, the associated Cauchy problem is ill-posed in the sense that uniform continuity with respect to the initial data fails. Thus, our result closes the discussion in the setting of the Sobolev spaces H s . The proof is achieved by combining the profile decomposition techniques with the integrability structure of the equation. Contents 1 2 potentials 2.3. Bäcklund transformation 3. Proof of the main theorem 3.1. Strategy of proof 3.2. Rigidity type results 3.3. End of the proof of Theorem 1 Appendix A. Regularized Determinants Appendix B. Proof of the estimate (2.74) References

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confidence: 99%

Global well-posedness for the derivative nonlinear Schrödinger equation

Bahouri¹,

Perelman²

2020

Preprint

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“…where ρ, λ k and C k are associated to the initial data. The proof of the following theorem can be found in [22,Section 3].…”

Section: The Direct and Inverse Mapsmentioning

confidence: 99%

The derivative nonlinear Schrödinger equation: Global well-posedness and soliton resolution

et al. 2019

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We review recent results on global wellposedness and long-time behavior of smooth solutions to the derivative nonlinear Schrödinger (DNLS) equation. Using the integrable character of DNLS, we show how the inverse scattering tools and the method of Zhou [48] for treating spectral singularities lead to global wellposedness for general initial conditions in the weighted Sobolev space H 2,2 pRq. For generic initial data that can support bright solitons but exclude spectral singularities, we prove the soliton resolution conjecture: the solution is asymptotic, at large times, to a sum of localized solitons and a dispersive component, Our results also show that soliton solutions of DNLS are asymptotically stable.

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“…A line of results, due to Hayashi-Ozawa [8], Colliander-Keel-Staffilani-Takaoka-Tao [4], Wu [28], and Guo-Wu [7], establishes global well-posedness of the DNLS equation in H s (R) for s ≥ 1 2 , for initial data having mass less than 4π. Another line (Pelinovsky-Saalmann-Shimabukuro [23], Pelinovsky-Shimabukuro [24], and Jenkins-Liu-Perry-Sulem [12,11,10]) uses inverse scattering techniques to establish global wellposedness under stronger regularity and decay assumptions on the initial data, but without a smallness requirement on the mass.…”

Section: Introductionmentioning

confidence: 99%

“…For noninteger s, we will use a sort of 'generalized energy', comparable to u 2 Ḣs (R) , that will be defined in terms of the transmission coefficient of the DNLS spectral problem. We sketch presently the background necessary to define these objects precisely; for more details, see, for example, [1,12,11,10,13,19,24,27].…”

Section: Introductionmentioning

confidence: 99%

$H^s$ Bounds for the Derivative Nonlinear Schrödinger Equation

Bahouri,

Leslie,

Perelman

2021

Preprint

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Global well-posedness for the derivative non-linear Schrödinger equation

Cited by 35 publications

References 21 publications

Global well-posedness for the derivative nonlinear Schrödinger equation

Global well-posedness for the derivative nonlinear Schrödinger equation

The derivative nonlinear Schrödinger equation: Global well-posedness and soliton resolution

$H^s$ Bounds for the Derivative Nonlinear Schrödinger Equation

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