On self-similar singular solutions of the complex Ginzburg-Landau equation

Communications on Pure and Applied Mathematics

Volumn 54, Issue 10, 2001, Pages 1215-1242

  Plecháč, Petr ^a   Šverák, Vladimír ^b  

^a UNIVERSITY OF WARWICK   (United Kingdom)

^b University of Minnesota   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0035613474     PISSN: 00103640     EISSN: None     Source Type: Journal    
DOI: 10.1002/cpa.3006     Document Type: Article

References (20)

1. Bartuccelli, M.; Constantin, P.; Doering, C. R.; Gibbon, J. D.; Gisselfält, M. On the possibility of soft and hard turbulence in the complex Ginzburg-Landau equation. Phys. D 44 (1990), no. 3, 421-444.
2. Budd, C. J.; Chen, S.; Russell, R. D. New self-similar solutions of the nonlinear Schrödinger equation with moving mesh computations. J. Comput. Phys. 152 (1999), no. 2, 756-789.
3. Doering, C. R.; Gibbon, J. D.; Levermore, C. D. Weak and strong solutions of the complex Ginzburg-Landau equation. Phys. D 71 (1994), no. 3, 285-318.
4. Klainerman, S. PDE as a unified subject. Proceeding of Visions in Mathematics. GAFA, 2000.
5. Kopell, N,; Landman, M. Spatial structure of the focusing singularity of the nonlinear Schrödinger equation: a geometrical analysis. SIAM J. Appl. Math. 55 (1995), no. 5, 1297-1323.
6. Kosmatov, N. E.; Petrov, I. V.; Shvets, V. F; Zakharov, V. E. Large amplitude simulation of wave collapse in nonlinear Schrödinger equations. Technical report, Academy of Science USSR, Space Research Institute, 1988.
7. Landman, M. J.; Papanicolaou, G. C.; Sulem, C.; Sulem, P.-L.; Wang, X. P. Stability of isotropic singularities for the nonlinear Schrödinger equation. Phys. D 47 (1991), no. 3, 393-415.
8. LeMesurier, B. J. The focusing singularity of the nonlinear Schrödinger equation. Doctoral dissertation, New York University, 1986.
9. LeMesurier, B. J.; Papanicolaou, G. C.; Sulem, C.; Sulem, P.-L. Local structure of the self-focusing singularity of the nonlinear Schrödinger equation. Phys. D 32 (1988), no. 2, 210-226.
10. Leray, J. Sur le mouvement d'un liquide visqueux emplissant l'espace. Acta Math. 63 (1934), 193-248.
11. Luke, Y. L. The special functions and their approximations. Mathematics in Science and Engineering, 53. Academic Press, New York-London, 1969.
12. McLaughlin, D. W.; Papanicolaou, G. C.; Sulem, C.; and Sulem, P.-L. Focusing singularity of the cubic Schrödinger equation. Phys. Rev. A 34 (1986), no. 2, 1200-1210.
13. Nečas, J.; Růzička, M.; Šverák, V. On Leray's self-similar solutions of the Navier-Stokes equations. Acta Math. 176 (1996), no. 2, 283-294.
14. Plecháč, P. Numerical analysis of Takens-Bogdanov points. Doctoral dissertation. Charles University, Prague, 1992.
15. Slater, L. J. Confluent hypergeometric functions. Cambridge University Press, New York, 1960.
16. Sulem, C.; Sulem, P.-L. The nonlinear Schrödinger equation. Self-focusing and wave collapse. Applied Mathematical Sciences, 139. Springer, New York, 1999.
17. Tsai, T.-P. On Leray's self-similar solutions of the Navier-Stokes equations satisfying local energy estimates. Arch. Rational Mech. Anal. 143 (1998), no. 1, 29-51.
18. Whittaker, E. T.; Watson, G. N. A course of modern analysis. Cambridge University Press, 1958.
19. Yan, X. Partial regularity of suitable weak solutions of complex Ginzburg Landau equations. Comm. Partial Differential Equations 24 (1999), no. 11-12, 2263-2277.
20. Zakharov, V. E. Collapse of self-focusing Langmuir waves. Handbook of plasma physics, 2. North Holland, Amsterdam, 1984.