Instability of the solitary wave solutions for the generalized derivative nonlinear Schrödinger equation in the endpoint case

Li, Bing; Ning, Cui

doi:10.48550/arxiv.1804.02738

Cited by 2 publications

(2 citation statements)

References 26 publications

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“…In the borderline case when c = 2z 0 √ ω and 1 < σ < 2, Fukaya ( [15], see also [22]) proved orbital instability of the solitons. This completes the study of orbital stability of the solitons of (gDNLS), except in the case of the algebraic soliton, which requires special attention [21,44].…”

Section: History On Well-posedness and Solitonsmentioning

confidence: 58%

Global well-posedness for the generalized derivative nonlinear Schrödinger equation

Pineau¹,

Taylor²

2021

Preprint

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In this article we study the well-posedness of the generalized derivative nonlinear Schrödinger equation (gDNLS)for small powers σ. We analyze this equation at both low and high regularity, and are able to establish global well-posedness in H s when s ∈ [1, 4σ) and σ ∈ (2 , 1). Our result when s = 1 is particularly relevant because it corresponds to the regularity of the energy for this problem. Moreover, a theorem of Liu, Simpson and Sulem (J. Nonlinear Sci. 2013) establishes the orbital stability of the gDNLS solitons, provided that there is a suitable H 1 well-posedness theory.To our knowledge, this is the first low regularity well-posedness result for a quasilinear dispersive model where the nonlinearity is both rough and lacks the decay necessary for global smoothing type estimates. These two features pose considerable difficulty when trying to apply standard tools for closing low-regularity estimates. While the tools developed in this article are used to study gDNLS, we believe that they should be applicable in the study of local well-posedness for other dispersive equations of a similar character. It should also be noted that the high regularity wellposedness presents a novel issue, as the roughness of the nonlinearity limits the potential regularity of solutions. Our high regularity well-posedness threshold s < 4σ is twice as high as one might naïvely expect, given that the function z → |z| 2σ is only C 1,2σ−1 Hölder continuous. Moreover, although we cannot prove H 1 well-posedness when σ ≤ √ 3 2 , we are able to establish H s well-posedness in the high regularity regime s ∈ (2 − σ, 4σ) for the full range of σ ∈ ( 1 2 , 1). This considerably improves the known local results, which had only been established in either H 2 or in weighted Sobolev spaces.

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Section: History On Well-posedness and Solitonsmentioning

confidence: 58%

Global well-posedness for the generalized derivative nonlinear Schrödinger equation

Pineau¹,

Taylor²

2021

Preprint

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“…For σ > 1, Fukaya, Hayashi and Inui [15] gave a sufficient condition of initial data for global well-posedness in H 1 (R). Some other results related to the stability theory and inverse scattering theory can be found in [7,8,16,18,20,37,39,44,45,50,51,52,53,54,55,63,64,69] and the references therein.…”

Section: Introductionmentioning

confidence: 95%

Optimal small data Scattering for the generalized derivative nonlinear Schrödinger equations

Bai,

Wu,

Xue

2018

Preprint

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In this work, we consider the following generalized derivative nonlinear Schrödinger equationsWe prove that when σ ≥ 2, the solution is global and scattering when the initial data is small in H s (R), s ≥ 1 2 . Moreover, we show that when 0 < σ < 2, there exist a class of solitary wave solutions {φ c } satisfywhen c tends to some endpoint, which is against the small data scattering statement. Therefore, the restriction σ ≥ 2 is optimal for scattering.

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Instability of the solitary wave solutions for the generalized derivative nonlinear Schrödinger equation in the endpoint case

Cited by 2 publications

References 26 publications

Global well-posedness for the generalized derivative nonlinear Schrödinger equation

Global well-posedness for the generalized derivative nonlinear Schrödinger equation

Optimal small data Scattering for the generalized derivative nonlinear Schrödinger equations

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