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THE GLOBAL CAUCHY PROBLEM FOR THE NLS WITH HIGHER ORDER ANISOTROPIC DISPERSION

Part of: Equations of mathematical physics and other areas of application Partial differential equations

Published online by Cambridge University Press:  12 December 2019

LEONID CHAICHENETS  and
NIKOLAOS PATTAKOS
LEONID CHAICHENETS
Affiliation:
Department of Mathematics, Institute for Analysis, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany e-mails: leonid.chaichenets@kit.edu; nikolaos.pattakos@kit.edu
NIKOLAOS PATTAKOS
Affiliation:
Department of Mathematics, Institute for Analysis, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany e-mails: leonid.chaichenets@kit.edu; nikolaos.pattakos@kit.edu
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Abstract

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We use a method developed by Strauss to obtain global well-posedness results in the mild sense and existence of asymptotic states for the small data Cauchy problem in modulation spaces ${M}^s_{p,q}(\mathbb{R}^d)$, where q = 1 and $s\geq0$ or $q\in(1,\infty]$ and $s>\frac{d}{q'}$ for a nonlinear Schrödinger equation with higher order anisotropic dispersion and algebraic nonlinearities.

MSC classification

Primary: 35A01: Existence problems 35A02: Uniqueness problems 35Q55: NLS-like equations (nonlinear Schrödinger)
Secondary: 35B40: Asymptotic behavior of solutions
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 63 , Issue 1 , January 2021 , pp. 45 - 53
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© Glasgow Mathematical Journal Trust 2019

References

REFERENCES

Ben-Artzi, M., Koch, H. and Saut, J.-C., Dispersion estimates for fourth order Schrödinger equations. C. R. Acad. Sci. Paris, Sér. I Math. 330 (2000), 87–92.Google Scholar
Bouchel, O., Remarks on NLS with higher order anisotropic dispersion. Adv. Differential Equ. 13(1–2) (2008), 169–198.Google Scholar
Carvajal, X., Panthee, M. and Scialom, M., On well-posedness of the third order NLS equation with time dependent coefficients. Commun. Contemp. Math. (2015). DOI: 10.1142/S021919971450031X.CrossRefGoogle Scholar
Chaichenets, L., Modulation spaces and nonlinear Schrödinger equations. PhD Thesis (Karlsruhe Institute of Technology, 2018). DOI: 10.5445/IR/1000088173.CrossRefGoogle Scholar
Chaichenets, L., Hundertmark, D., Kunstmann, P. and Pattakos, N., Local well-posedness for the nonlinear Schrödinger equation in modulation spaces ${M}^s_{p,q}(\mathbb{R}^d)$ . preprint (2016), arXiv:1610.08298.Google Scholar
Diaz-Ortero, F. J. and Chamorro-Posada, P., Interchannel soliton collisions in periodic dispersion maps on the presence of third order dispersion. J. Nonlinear Math. Phys. 15 (2008), 137–143.Google Scholar
Feichtinger, H. G., Modulation spaces on locally compact Abelian groups. Technical Report, University of Vienna, 1983, in Proceedings of the International Conference on Wavelet and Applications (New Delhi Allied Publishers, India, 2003), 99–140.Google Scholar
Karlsson, M. and Höök, A., Soliton-like pulses governed by fourth order dispersion in optical fibers. Opt. Commun. 104 (1994), 303–307.Google Scholar
Karpman, V.I., Stabilization of soliton instabilities by higher-order dispersion: Fourth order nonlinear Schrödinger-type equations. Phys. Rev. E 53(2) (1996), R1336–R1339.Google ScholarPubMed
Kato, T., The global Cauchy problems for the nonlinear dispersive equations on modulation spaces. J. Math. Anal. Appl. 413 (2014), 821–840.CrossRefGoogle Scholar
Kato, T., Solutions to nonlinear higher order Schrödinger equations with small initial data on modulation spaces. Adv. Differential Equations 21 (2016), 201–234.Google Scholar
Kunstmann, P., Modulation type spaces for generators of polynomially bounded groups and Schrödinger equations. Semigroup Forum, (2019). DOI: 10.1007/s00233-019-10016-1.CrossRefGoogle Scholar
Ruzhansky, M. and Ville, T., Pseudo-Differential Operators and Symmetries, vol. 2, Pseudo-Differential Operators (Birkhäuser, Basel, 2010), ISBN 978-3-7643-8513-2.CrossRefGoogle Scholar
Saut, J.-C. and Segata, J.-I., Asymptotic behavior in time of solution to the nonlinear Schrödinger equation with higher order anisotropic dispersion. Discrete Contin. Dyn. Syst. 39 (2019), 219–239.CrossRefGoogle Scholar
Strauss, W., Nonlinear scattering theory at low energy. J. Funct. Anal. 41 (1981), 110–133.CrossRefGoogle Scholar
Sugimoto, M., Tomita, N. and Wang, B., Remarks on nonlinear operations on modulation spaces. Integr. Transf. Spec. F. 22(4–5) (2011), 351–358.CrossRefGoogle Scholar
Wang, B. X. and Hudzik, H., The global Cauchy problem for the NLS and NLKG with small rough data. J. Differ. Eq. 232 (2007), 36–73.CrossRefGoogle Scholar