Publication Cover
Applicable Analysis
An International Journal
Volume 88, 2009 - Issue 1
67
Views
2
CrossRef citations to date
0
Altmetric
Original Articles

Instability of standing waves for a weakly coupled non-linear Schrödinger system

&
Pages 53-62 | Received 08 Jul 2008, Accepted 18 Sep 2008, Published online: 21 May 2009

References

  • Berge , L . 1998 . Wave collapse in physics: principles and applications to light and plasma waves . Phys. Rep. , 303 : 259 – 370 .
  • Fibich , G . 2001 . Self-focusing in the perturbed and unperturbed nonlinear Schrödinger equation in critical dimension . SIAM J. Appl. Math. , 61 : 1680 – 1705 .
  • Lakshmanan , M , Kanna , T and Radhakrishnan , R . 2000 . Shape-changing collisions of coupled bright solitons in birifringent optical fibers . Rep. Math. Phys. , 46 : 143 – 156 .
  • Menyuk , CR . 1987 . Nonlinear pulse propagation in birifringent optical fibers . IEEE J. Quantum Electron. , 23 : 174 – 176 .
  • Ginibre , J and Velo , G . 1979 . On a class of nonlinear 1680–1705 Schrödinger equations . J. Funct. Anal. , 32 : 1 – 71 .
  • Kato , T . 1987 . On nonlinear Schrödinger equations . Ann. Inst. Henri Poincaré, Physique Théorique , 49 : 113 – 129 .
  • Glassey , RT . 1977 . On the blowup of nonlinear Schrödinger equations . J. Math. Phys. , 18 ( 9 ) : 1794 – 1797 .
  • Ogawa , T and Tsutsumi , Y . 1991 . Blow-up of H 1 solution for the nonlinear Schrödinger equation . J. Diff. Eqns. , 92 ( 2 ) : 317 – 330 .
  • Ogawa , T and Tsutsumi , Y . 1991 . Blow-up of H 1 solution for the nonlinear Schrödinger equation with critic power nonlinearity . Proc. Amer. Math. Soc. , 111 ( 2 ) : 487 – 496 .
  • Berestycki , H and Cazenave , T . 1981 . Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéarires . C.R. Acad. Sci. Paris, Seire I , 293 : 489 – 492 .
  • Weinstein , MI . 1983 . Nonlinear Schrödinger equations and sharp interpolations estimates . Comm. Math. Phys. , 87 : 567 – 576 .
  • Zhang , J . 2002 . Sharp conditions of global existence for nonlinear Schrödinger and Klein-Gordon equations . Nonlinear Analysis , 48 : 191 – 207 .
  • Zhang , J . 2001 . “ Proceedings of the Smalefest ” . In Cross-constrained variational problem and nonlinear Schrödinger equations , 457 – 469 . Hong Kong : Foundation of Computational Mathematics .
  • Cazenave , T . 1989 . “ Textos de Metodos Matematicos ” . In An introduction to nonlinear Schrödinger equations , 48 – 73 . Riede Janerro .
  • Tsutsumi , Y and Zhang , J . 1998 . Instability of optical solitions for two-wave interaction model in cubic nonlinear media . Adv. Math. Sci. Appl. , 8 : 691 – 713 .
  • Fanelli , L and Montefusco , E . 2007 . On the blow-up threshold for two coupled nonlinear Schrödinger equations . J. Phys. A: Math. Theor. , 40 : 14139 – 14150 .
  • Ambrosetti , A and Colorado , E . 2007 . Standing waves of some coupled nonlinear Schrödinger equations . J. London Math. Soc. , 75 : 67 – 82 .
  • Maia , LA , Montefusco , E and Pellacci , B . 2006 . Positive solutions for a weakly coupled nonlinear Schrödinger system . J. Diff. Eqs. , 229 : 743 – 767 .
  • Payne , LE and Sattinger , DH . 1975 . Saddle points and instability of nonlinear hyperbolic equations . Israel J. Math. , 22 : 273 – 303 .
  • Levine , HA . 1974 . Instability and non-existence of global solutions to nonlinear wave equations of the form Pu tt = −Au + F(u) . Trans. Amer. Math. Soc , 192 : 1 – 21 .
  • Berestycki , H and Lions , PL . 1983 . Nonlinear scalar field equations, I. Existence of a ground state . Arch. Ration. Mech. Anal. , 82 : 313 – 345 .
  • Strauss , WA . 1977 . Existence of solitary waves in high dimensions . Commun. Math. Phys. , 55 : 149 – 162 .

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.