Hostname: page-component-cb9f654ff-hn9fh Total loading time: 0 Render date: 2025-08-30T15:27:23.991Z Has data issue: false hasContentIssue false
  • English
  • Français

ON GLOBAL ROUGH SOLUTIONS TO A NON-LINEAR SCHRÖDINGER SYSTEM

Published online by Cambridge University Press:  01 September 2009

LI MA ,
XIANFA SONG  and
LIN ZHAO
LI MA
Affiliation:
Department of Mathematical Sciences, Tsinghua University, Peking 100084, P.R. China e-mail: lma@math.tsinghua.edu.cn
XIANFA SONG
Affiliation:
Department of Mathematics, Tianjin University, Tianjin 300072, P.R. China e-mail: songxianfa2004@163.com
LIN ZHAO
Affiliation:
Department of Mathematical Sciences, Tsinghua University, Beijing, P.R. China e-mail: zhaolin05@mails.tsinghua.edu.cn
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The non-linear Schrödinger systems arise from many important physical branches. In this paper, employing the ‘I-method’, we prove the global-in-time well-posedness for a coupled non-linear Schrödinger system in Hs(n) when n = 2, s > 4/7 and n = 3, s > 5/6, respectively, which extends the results of J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao (Almost conservation laws and global rough solutions to a nonlinear Schrödinger equation, Math Res. Lett. 9, 2002, 659–682) to the system.

Keywords

Primary 35Q55

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 51 , Issue 3 , September 2009 , pp. 499 - 511
Copyright
Copyright © Glasgow Mathematical Journal Trust 2009

References

REFERENCES

1.Akhmediev, N. and Ankiewicz, A., Partially coherent solitons on a finite background, Phys. Rev. Lett. 82 (1999), 2661.CrossRefGoogle Scholar
2.Bourgain, J., Refinements of Strichartz's inequality and applications to 2D-NLS with critical nonlinearity, Int. Math. Res. Notices 5 (1998), 253–283.CrossRefGoogle Scholar
3.Bourgain, J., Global solutions of nonlinear Schrödinger equations (American Mathematical Society, Providence, RI, 1999).CrossRefGoogle Scholar
4.Buljan, H., Schwartz, T., Segev, M., Soljacic, M. and Christoudoulides, D., Polychromatic partially spatially incoherent solitons in a noninstantaneous Kerr nonlinear medium, J. Opt. Soc. Am. B 21 (2004), 397–404.CrossRefGoogle Scholar
5.Cazenave, T. and Weissler, F., The Cauchy problem for the nonlinear Schrödinger equation in H 1, Manuscripta Math. 61 (1988), 477–494.CrossRefGoogle Scholar
6.Colliander, J., Keel, M., Staffilani, G., Takaoka, H. and Tao, T., Almost conservation laws and global rough solutions to a nonlinear Schrödinger equation, Math. Res. Lett. 9 (2002), 659–682.CrossRefGoogle Scholar
7.Colliander, J., Keel, M., Staffilani, G., Takaoka, H. and Tao, T., Multi-linear estimates for periodic Kdv equations, and applications, J. Funct. Anal. 211 (2004), 173–218.CrossRefGoogle Scholar
8.Colliander, J., Raynor, S., Sulem, C. and Wright, J. D., Ground state mass concentration in the L 2-critical nonlinear Schrödinger equation below H 1, Math. Res. Lett. 12 (2005), 357–375.CrossRefGoogle Scholar
9.Gidas, B., Ni, W. M. and Nirenberg, L., Symmetry of positive solutions of nonlinear elliptic equations in n, Adv. Math. Studies 7 (1981), 369–402.Google Scholar
10.Hioe, F. T., Solitary waves for N coupled nonlinear Schrödinger equations, Phys. Rev. Lett. 82 (1999), 1152–1155.CrossRefGoogle Scholar
11.Ma, L. and Zhao, L., Sharp thresholds of blow-up and global existence for the coupled nonlinear Schrödinger system, J. Math. Phys. 49 (2008), 062103.CrossRefGoogle Scholar
12.Ma, L. and Zhao, L., Uniqueness of ground state of some coupled nonlinear Schrödinger system, J. Diff. Eq. 245 (2008), 2551–2565.CrossRefGoogle Scholar
13.Ma, L. and Zhao, L., On energy stability for the coupled nonlinear Schrödinger system (Zeitschrift fur Angewandte Mathematik und Physik, 2009).CrossRefGoogle Scholar
14.Takaoka, H., Global well-posedness for the Schrödinger equations with derivative in a nonlinear term and data in low order Sobolev space, Electronic J. Diff. Eq. 42 (2001), 1–23.Google Scholar
15.Yajima, K., Existence of solutions for Schrödinger evolution equations, Comm. Math. Phys. 110 (1987), 415–426.CrossRefGoogle Scholar