Blow-up of spatially periodic complex-valued solutions of nonlinear dispersive equations

Indiana University Mathematics Journal

Volumn 50, Issue 2, 2001, Pages 759-782

  Bona, J L ^a,b   Weissler, F B ^c  

^a UNIVERSITY OF TEXAS AT AUSTIN   (United States)

^b UNIVERSITY OF ILLINOIS AT CHICAGO   (United States)

^c UNIVERSITÉ PARIS 13   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0039838728     PISSN: 00222518     EISSN: None     Source Type: Journal    
DOI: 10.1512/iumj.2001.50.1865     Document Type: Article

References (46)

1. G. AKRIVIS, V.A. DOUGALIS, O.A. KARAKASHIAN & W.R. MCKINNEY, Numerical approximations of blow-up of radially symmetric solutions of the nonlinear Schro"dinger equation, SIAM J. Sci. Comp, to appear.
2. J.P. ALBERT, J.L. BONA & M. FELLAND A criterion for the formation of singularities for the generalized Korteweg-de Vries equation Matemática Aplicada e Computacional 7 (1988), 3-11.
3. J. ANGULO, J.L. BONA, F. LINARES & M. SCIALOM, On the structure of singularities in solutions of the nonlinear Schrödinger equation for the critical case, p = 4/n, In: Nonlinear Theory of Generalized Functions (M. Grosser, G. Hoermann, M. Kunzinger & M. Oberguggenberger, eds.), Chapman & Hall, Boca Raton, 1999, pp. 3-22.
4. _, Scaling, stability and singularities for nonlinear dispersive wave equations: The critical case, Nonlinearity (to appear).
5. E.D. BELOKOLOS, A.I. BOBENKO, V.Z. ENOL'SKII & A.R. ITS, Algebro-geometric approach to nonlinear integrable equations, Springer Series in Nonlinear Dynamics, Springer-Verlag, New York-Heidelberg-Berlin, 1994.
6. B. BIRNIR, Singularities of the complex Korteweg-de Vries flows, Commun. Pure Appl. Math. 39 (1986), 283-305.
7. _, An example of blow-up, for the complex KdV equation and existence beyond the blow-up, SIAM J. Math. Anal. 47 (1987), 710-725.
8. J.L. BONA, V.A. DOUGALIS & O.A. KARAKASHIAN, Fully discrete Galerkin methods for the Korteweg-de Vries equation, J. Comp. and Math. with Appl. 12A (1986), 859-884.
9. J.L. BONA, V.A. DOUGALIS, O.A. KARAKASHIAN & W.R. MCKINNEY, Computations of blow up and decay for periodic solutions of the generalized Korteweg-de Vries equation, Applied Numerical Mathematics 10 (1992), 335-355.
10. _, Conservative, high-order numerical schemes for the generalized Korteweg-de Vries equation, Philos. Trans. Royal Soc. London, Series A 351 (1995), 107-164.
11. _, Numerical simulation of singular solutions of the generalized Korteweg-de Vries equation, In the Volume on Hydrodynamic Waves (F. Dias, J.-M. Ghidaglia & J.-C. Saut, eds.), Contemporary Math. 200 (1996), 17-29.
12. J.L. BONA & J.-C. SAUT, Dispersive blow-up of solutions of generalized Korteweg-de Vries equations, J. Diff. Eq. 103 (1993), 3-57.
13. J.L. BONA, P.E. SOUGANIDIS & W.A. STRAUSS, Stability and instability of solitary waves of KdV-type, Proc. Royal Soc. London, Series A 411 (1993), 395-412.
14. J.L. BONA & F.B. WEISSLER, Similarity solutions of the generalized Korteweg-de Vries equation, Math. Proc. Cambridge Philos. Soc. 127 (1999), 323-351.
15. T. CAZENAVE, An introduction to nonlinear Schrödinger equations, Textos de Métodos Matemáticos 26, Universidade Federal do Rio de Janeiro: Rio de Janeiro, 2nd Ed. (1993).
16. _, Blow-up and scattering in the nonlinear Schrödinger equation, Textos de Mérodos Matemáticos 30, Universidade Federal do Rio de Janeiro: Rio de Janeiro (1994).
17. T. CAZENAVE & F.B. WEISSLER, The structure of solutions to the pseudo-conformally invariant nonlinear Schrödinger equation, Proc. Royal Soc. Edinburgh 117A (1991), 251-273.
18. W. CRAIG, T. KAPPELER AND W. STRAUSS, Infinite gain of regularity for equations of KdV type, Ann. l'Inst. Henri Poincaré, "Analyse Nonlinéaire" 9 (1992), 147-186.
19. D. DIX, The dissipation of nonlinear dispersive waves: The case of asymptotically weak nonlinearity, Commun. P.D.E. 17 (1992), 1655-1693.
20. _, Large-time asymptotic behavior of solutions of linear dispersive equations, Springer Lecture Notes in Mathematics 1668 (1997).
21. R. GLASSEY, On the blowing up of solutions of the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys. 18 (1977), 1794-1797.
22. O. KAVIAN, A remark on the blowing-up of solutions to the Cauchy problem for nonlinear Schrödinger equations, Trans. American Math. Soc. 299 (1987), 193-205.
23. E.W. LAEDKE, L. BLAHA & K.H. SPATSCHEK, Collapsing states of generalized Korteweg-de Vries equations, Physica D 40 (1989), 249-264.
24. E.W. LAEDKE, L. BLAHA, K.H. SPATSCHEK AND E.A. KUZNETSOV, On stability of collapse in the critical case, J. Math. Phys. 33 (1992), 967-973.
25. E.W. LAEDKE AND K.H. SPATSCHEK, Variational principles in soliton physics, In: Diff. Geometry, Calculus of Variations and their Application, Lecture Notes in Pure and Appl. Math. Volume 100, Dekker, New York, 1985, pp. 335-357.
26. E.W. LAEDKE, K.H. SPATSCHEK & L. STENFLO, Evolution theorem for a class of perturbed envelope soliton solutions, J. Math. Phys. 24 (1983), 2764-2769.
27. M.J. LANDMAN, G. PAPANICOLAOU, C. SULEM & P.L. SULEM, Rate of blow-up for solutions of the nonlinear Schrödinger equation, Phys. Rev. A 38 (1988), 3837-3843.
28. B. LE MESURIER, G. PAPANICOLAOU, C. SULEM AND P.-L. SULEM, Focusing and multi-focusing solutions of the nonlinear Schrödinger equation, Physica D 31 (1988), 78-102.
29. _, Local structure of the self-focusing singularity of the nonlinear Schrödinger equation, Physica D 32 (1988), 210-266.
30. Y. MARTEL, A nonlinear Airy equation, Comp. and Appl. Math. 15 (1996), 1-17.
31. _, Blow up for the nonlinear Schrödinger equation in non-isotropic spaces, Nonlinear Anal. TMA (to appear).
32. Y. MARTEL AND F. MERLE, Stability of blowup profile and lower bounds on the blowup rate for the critical generalized KdVequation, Annals Math. (to appear).
33. _, Instability of solitons for the critical generalized Korteweg-de Vries equation, Geom. Funct. Anal. 11 (2001), 74-123.
34. F. MERLE, Construction of solutions with exactly k blow-up points for the nonlinear Schrödinger equation with critical nonlinearity, Commun. Math. Phys. 129 (1990), 223-240.
35. _, Determination of blow-up solutions with minimal mass for the nonlinear Schrödinger equation with critical power, Duke Math. J. 69 (1993), 427-454.
36. _, Existence of blowup solutions in the energy space for the critical generalized KdV equation, J. Amer. Math. Soc. (to appear).
37. 1 solutions for the nonlinear Schrödinger equation, J. Diff. Eq. 92 (1991), 317-330.
38. 1 solutions for the one-dimensional Schrödinger equation with critical power nonlinearity, Proc. American Math. Soc. 111 (1991), 487-496.
39. _, Blow-up of solutions for the nonlinear Schrödinger equation with quartic potential and periodic boundary conditions, Functional Analytic methods for partial Differential Equations (H. Fujita, T. Ikebe & S.T. Kutoda eds.), Springer Lecture Notes Math. 1450 (1990), 236-261.
40. C. SULEM & P.L. SULEM, Nonlinear Schrödinger equation: self-focusing and wave collapse, in preparation.
41. Y. TOURIGNY & J.M. SANZ-SERNA, The numerical study of blow-up with an application to nonlinear Schrödinger equation, J. Comp. Phys. 102 (1992), 407-416.
42. F. TREVES, Topological Vector Spaces, Distributions and Kernels, Academic Press: New York (1967).
43. M.I. WEINSTEIN, On the structure and formation of singularities to nonlinear dispersive equations, Commun. P.D.E. 11 (1986), 545-565.
44. _, Existence and dynamical stability of solitary wave solutions of equations arising in long wave propagation, Commun. P.D.E. 12 (1987), 1133-1173.
45. _, The nonlinear Schrödinger equation - singularity formation, stability and dispersion, In: The Connection Between Infinite-dimensional and Finite-dimensional Systems (B. Nicolaenko, C. Foias & R. Temam, eds.), American Math. Soc., Providence, 1989, pp. 213-232.
46. V.E. ZAKHAROV, Collapse of Langmuir waves, Soviet Phys. JETP 35 (1972), 908-914.