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Abstract This article is devoted to a general class of one-dimensional NLS problems with a cubic nonlinearity. The question of obtaining scattering, global in time solutions for such problems has attracted a lot of attention in recent years, and many global well-posedness results have been proved for a number of models under the assumption that the initial data are bothsmallandlocalized. However, except for the completely integrable case, no such results have been known for small but not necessarily localized initial data. In this article, we introduce a new, nonperturbative method to prove global well-posedness and scattering for$$L^2$$initial data which aresmallandnonlocalized. Our main structural assumption is that our nonlinearity isdefocusing. However, we do not assume that our problem has any exact conservation laws. Our method is based on a robust reinterpretation of the idea of Interaction Morawetz estimates, developed almost 20 years ago by the I-team. In terms of scattering, we prove that our global solutions satisfy both global$$L^6$$Strichartz estimates and bilinear$$L^2$$bounds. This is a Galilean invariant result, which is new even for the classical defocusing cubic NLS.1There, by scaling, our result also admits a large data counterpart.
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Long Time Solutions for 1D Cubic Dispersive Equations, Part II: The Focusing Case
This article is concerned with one dimensional dispersive flows with cubic non- linearities on the real line. In a very recent work, the authors have introduced a broad conjecture for such flows, asserting that in the defocusing case, small initial data yields global, scattering solutions. Then this conjecture was proved in the case of a Schr¨odinger dispersion relation. In terms of scattering, our global solutions were proved to satisfy both global L6 Strichartz estimates and bilinear L2 bounds. Notably, no localization assumption is made on the initial data. In this article we consider the focusing scenario. There potentially one may have small solitons, so one cannot hope to have global scattering solutions in general. Instead, we look for long time solutions, and ask what is the time-scale on which the solutions exist and satisfy good dispersive estimates. Our main result, which also applies in the case of the Schr¨odinger dispersion relation, asserts that for initial data of size ϵ, the solutions exist on the time-scale ϵ^{-8}, and satisfy the desired L^6 Strichartz estimates and bilinear L^2 bounds on the time-scale ϵ^{−6}. To the best of our knowledge, this is the first result to reach such a threshold.
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- PAR ID:
- 10581257
- Publisher / Repository:
- Springer Singapore Vietnam. Math. Soc. Vietnam. Acad. Sci. Technol., Inst. Math.
- Date Published:
- Journal Name:
- Vietnam Journal of Mathematics
- Volume:
- 52
- Issue:
- 3
- ISSN:
- 2305-221X
- Page Range / eLocation ID:
- 597 to 614
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s10013-023-00660-0
