Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-04T19:12:19.570Z Has data issue: false hasContentIssue false
  • English
  • Français

Equations de Schrödinger non linéaires en dimension deux

Published online by Cambridge University Press:  14 November 2011

Thierry Cazenave
Thierry Cazenave
Affiliation:
Université Pierre et Marie Curie, Laboratoire Analyse Numerique, 4 place Jussieu, Paris, France
Get access Rights & Permissions [Opens in a new window]

Synopsis

This paper is devoted to the study of some non linear Schrödinger equations in two dimensions, arising in non linear optics; in particular, it is concerned with solutions to the Cauchy problem. The problem of global existence and regularity of the solutions, the asymptotic behaviour of global solutions, and the blow-up of non global solutions are studied.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 84 , Issue 3-4 , 1979 , pp. 327 - 346
Copyright
Copyright © Royal Society of Edinburgh 1979

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Blbliographie

1Aitken, A. H., Hayes, J. N., Ulrich, P. B.. Propagation of high energy 10–6 μ laser beams through the atmosphere. N.R.L. Report 7293 (1971).Google Scholar
2Amann, H.. Periodic solutions of semi-linear parabolic equations (Preprint).Google Scholar
3Baillon, J. B., Cazenave, T., Figueira, M.. Equation de Schrödinger non linéaire. C.R. Acad. Sci. Paris Sér. A-B 284 (1977), 869–872.Google Scholar
4Baillon, J. B., Cazenave, T., Figueira, M.. Equation de Schrödinger avec non-linéarité intégrate. C. R. Acad. Sci. Paris Sér. A-B 284 (1977), 939–942.Google Scholar
5.Berestycki, H., Lions, P. L.. Existence d'ondes solitaires dans les problèmes non-linéaires du type Klein-Gordon. C. R. Acad. Sci. Paris Sér. A-B 287 (1978), 503–506.Google Scholar
6Ginibre, J., Velo, G.. On a class of non-linear Schrödinger equations, I, II. J. Functional Analysis 32 (1979), 1–71; III, Ann. Inst. H. Poincaré 28 (1978), 287–316.CrossRefGoogle Scholar
7Glassey, R. T.. On the blowing-up of solutions to the Cauchy problem for the non linear Schrödinger equation (Preprint).Google Scholar
8Kelley, P. L.. Self focusing of optical beams. Phys. Rev. Lett 15 (1965), 1005–1008.CrossRefGoogle Scholar
9Lam, J. F., Lippmann, B., Tappert, F.. Self trapped laser beams in plasma. Phys. Fluids 20 (1977), 1176–1179.CrossRefGoogle Scholar
10Lin, J. E., Strauss, W. A.. Decay and scattering of solutions of a non linear Schrödinger equation (Preprint).Google Scholar
11Nirenberg, L.. On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa 13 (1959), 115–162.Google Scholar
12Reed, M., Simon, B.. Methods of Modern Mathematical Physics, II (New York: Academic Press, 1975).Google Scholar
13Segal, I.. Non linear semi-groups. Ann. Math. 78 (1963), 339–364.CrossRefGoogle Scholar
14Strauss, W. A.. Non linear invariant wave equations. Springer, à paraître.Google Scholar
15Strauss, W. A.. The non linear Schrödinger equation. In Contemporary Developments in Continuum Mechanics and Partial Differential Equations, p. 452–465 (ed. Penha-Medelros, De La) (Amsterdam: North-Holland, 1978).Google Scholar
16Suydam, B. R.. Self Focusing of very powerful laser beams. Spec. Publs Natn. Bur. Stand. 387 (1973), 42–48.Google Scholar