Asymptotic and Limiting Profiles of Blowup Solutions of the Nonlinear Schrödinger Equation with Critical Power

Communications on Pure and Applied Mathematics

Volumn 52, Issue 2, 1999, Pages 193-270

  Nawa, Hayato ^a  

^a NAGOYA UNIVERSITY   (Japan)

Author keywords

[No Author keywords available]

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EID: 0033484486     PISSN: 00103640     EISSN: None     Source Type: Journal    
DOI: 10.1002/(SICI)1097-0312(199902)52:2<193::AID-CPA2>3.0.CO;2-3     Document Type: Article

References (67)

1. Akhmanov, S. A., Sukhorukov, A. P., and Khokhlov, R. V., Self-focusing and self-trapping of intense light beams in a nonlinear medium, Soviet Phys. JETP 23, 1966, pp. 1025-1033.
2. Akhmanov, S. A., Sukhorukov, A. P., and Khokhlov, R. V., Self-focusing and diffraction of light in a nonlinear medium, Soviet Phys. USPEKHI 93, 1968, pp. 609-633.
3. Ball, J. M., Finite time blowup in nonlinear problems, pp. 189-205 in: Nonlinear Evolution Equations (Proc. Sympos., Univ. Wisconsin, Madison, Wis., 1977), Publ. Math. Res. Center Univ. Wisconsin No. 40, M. G. Crandall, ed., Academic Press, New York-London, 1978.
4. Berestycki, H., and Lions, P.-L., Nonlinear scalar field equations I, II, Arch. Rational Mech. Anal. 82, 1983, pp. 313-375.
5. Brézis, H., and Coron, J.-M., Convergence of solutions of H-system or how to blow bubbles, Arch. Rational Mech. Anal. 89, 1985, pp. 21-56.
6. Brézis, H., and Lieb, E., A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88, 1983, pp. s 486-490.
7. Brézis, H., and Lieb, E. H., Minimum action solutions of some vector field equations, Comm. Math. Phys. 96, 1984, pp. s 97-113.
8. s, Nonlinear Anal. 14, 1990, pp. 807-836.
9. Cazenave, T., and Weissler F. B., The structure of solutions to the pseudo-conformally invariaint nonlinear Schrödinger equations, Proc. Roy. Soc. Edinburgh Section A 117, 1991, pp. 251-273.
10. Chen, X.-Y., and Matano, H., Convergence, asymptotic periodicity, and finite-point blowup in one-dimensional semilinear heat equations, J. Differential Equations 78, 1989, pp. 160-190.
11. Fibich, G., and Papanicolaou, G., Self-focusing in the perturbed and unperturbed nonlinear Schrödinger equation in critical dimension, SIAM J. Appl. Math. 1997, submitted.
12. Fröhlich, J., Lieb, E. H., and Loss, M., Stability of Coulomb systems with magnetic fields I: The one-electron atom, Comm. Math. Phys. 104, 1986, pp. 251-270.
13. Giga, Y., and Kohn, R. V., Nondegeneracy of blowup for semilinear heat equations, Comm. Pure Appl. Math. 42, 1989, pp. 845-884.
14. Ginibre, J., and Velo, G., On a class of nonlinear Schrödinger equations I, II, J. Funct. Anal. 32, 1979, pp. 1-71.
15. Ginibre, J., and Velo, G., Sur une équation de Schrödinger non linéaire avec interaction non local, pp. 155-199, 391-392 in: Nonlinear Partial Differential Equations and Their Applications, Collège de France Seminar Vol. II (Paris, 1979/1980), H. Brézis, and P. L. Lions, eds., Res. Notes in Math. No. 60, Pitman, Boston, Mass.-London, 1982.
16. Ginibre, J., and Velo, G., The global Cauchy problem for the nonlinear Schrödinger equation revisited, Ann. Inst. H. Poincaré, Anal. Non Linéaire 2, 1985, pp. 309-327.
17. Glassey, R. T., On the blowing up solution to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys. 18, 1977, pp. 1794-1797.
18. Kato, T., On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré, Phys. Théor. 46, 1987, pp. 113-129.
19. Kato, T., Nonlinear Schrödinger equations, pp. 218-236 in: Schrödinger Operators (Søonderborg, 1988), H. Holden and A. Jensen, eds., Lecture Notes in Phys. No. 345, Springer-Verlag, Berlin, 1989.
20. n, Arch. Rational Mech. Anal. 105, 1989, pp. 243-266.
21. Landman, M. J., Papanicolaou, G. C., Sulem, C., Sulem, P.-L., and Wang, X. P., Stability of Isotropie singularities for the nonlinear Schrödinger equation, Phys. D 47, 1991, pp. 393-415.
22. LeMesurier, B., Papanicolaou, G., Sulem, C., and Sulem, P.-L., The focusing singularity of the nonlinear Schrödinger equation, pp. 159-201 in: Directions in Partial Differential Equations (Madison, WI, 1985), M. G. Crandall, P. H. Rabinowitz, and R. E. Turner, eds., Publ. Math. Res. Center Univ. Wisconsin No. 54, Academic Press, Boston, Mass., 1987.
23. Lieb, E. H., On the lowest eigenvalue of the Laplacian for the intersection of two domains, Invent. Math. 74, 1983, pp. 441-448.
24. Lions, P.-L., The concentration compactness principle in the calculus of variations. The locally compact case I, II, Ann. Inst. H. Poincaré, Anal. Non Linéaire 1, 1984, pp. 109-145, 223-283.
25. Lions, P. L., The concentration compactness principle in the calculas of variations. The limit case I, II, Riv. Math. Iberoamericana 1, 1985, pp. 145-201, 45-121.
26. McLaughlin, D. W., Papanicolaou, G., Sulem, C., and Sulem, P.-L., Focusing singularity of the cubic Schrödinger equation, Phys. Rev. A 34, 1986, pp. 1200-1210.
27. Merle, F., Construction of solutions with exactly k blowup points for the Schrödinger equation with critical power nonlinearity, Comm. Math. Phys. 129, 1990, pp. 223-240.
28. Merle, F., On uniqueness and continuation properties after blowup time of self-similar solutions of the nonlinear Schrödinger equation with critical exponent and critical mass, Comm. Pure Appl. Math. 45, 1992, pp. 203-254.
29. Merle, F., Determination of blowup solutions with minimal mass for nonlinear Schrödinger equations with critical power, Duke Math. J. 69, 1993, pp. 421-454.
30. Merle, F., Lower bounds for the blowup rate of solutions of the Zakharov equation in dimension two, Comm. Pure Appl. Math. 49(8), 1996, pp. 765-794.
31. 2 concentration of blowup solutions for the nonlinear Schrödinger equation with the critical power nonlinearity, J. Differential Equations 84, 1990, pp. 205-214.
32. Nagasaki, K., and Suzuki, T., Asymptotic analysis for two-dimensional elliptic eigenvalue problems with exponentially-dominated nonlinearities, Asymptotic Anal. 3, 1990, pp. 173-188.
33. Nawa, H., Blowup solutions for the nonlinear Schrödinger equations, 1986, unpublished manuscript.
34. Nawa, H., "Mass concentration" phenomenon for the nonlinear Schrödinger equation with the critical power nonlinearity II, Kodai Math. J. 13, 1990, pp. 333-348.
35. Nawa, H., Convergence theorems for the pseodo-conformally invariant nonlinear Schrödinger equation, Proc. Japan Acad. Ser. A Math. Sci. 66, 1990, pp. 214-216.
36. Nawa, H., Formation of singularities in solutions of the nonlinaer Schrödinger equation, Proc. Japan Acad. Ser. A Math. Sci. 67, 1991, pp. 29-34.
37. Nawa, H., "Mass concentration" phenomenon for the nonlinear Schrödinger equation with the critical power nonlinearity, Funkcial. Ekvac. 35, 1992, pp. 1-18.
38. Nawa, H., Asymptotic profiles of blowup solutions of the nonlinear Schrödinger equation, pp. 221-253 in: Singularities in Fluids, Plasmas and Optics, R. E. Caflisch, and G. C. Papanicolaou, eds., NATO ASI series Kluwer, Dordrecht, 1993. Revised version in Tokyo Institute of Technology preprint series NO. 06-93 (#15), 1993.
39. Nawa, H., Asymptotic profiles of blowup solutions of the nonlinear Schrödinger equation with critical power nonlinearity, J. Math. Soc. Japan 46, 1994, pp. 557-586.
40. Nawa, H., Asymptotic behavior of blowup solutions of the nonlinear Schrödinger equation with critical power nonlinearity, pp. 133-142 in: Proceeding of the Second International Colloquim on Numerical Analysis (Plovdiv, 1993), D. Bainov, and V. Covachev, eds., VSP, Utrecht, 1994.
41. Nawa, H., Formation of singularities in solutions of the nonlinear Schrödinger equation with critical power nonlinearity, pp. 167-188 in: Miniconference on Analysis and Applications (Brisbane, 1993), Proc. Centre Math. Appl. Austral. Nat. Univ. No. 33, G. Martin, and B. Thompson, eds., Australian National University, Canberra, 1994.
42. 1, preprint.
43. Nawa, H., Asymptotic profiles of blowup solutions of the nonlinear Schrödinger equation with critical power nonlinearity II, 1997, unpublished manuscript.
44. Nawa, H., Asymptotic profiles of blowup solutions of the nonlinear Schrödinger equation with critical power nonlinearity III, 1997, unpublished manuscript.
45. Nawa, H., Limiting profiles of blowup solutions of the nonlinear Schrödinger equation with critical power nonlinearity, Proc. Japan Acad. 73 (A), 1997, pp. 171-175.
46. Nawa, H., Single point blowup in solutions of the nonlinear Schrödinger equation with critical power nonlinearity, 1997, preprint.
47. Nawa, H., Two points blowup in solutions of the nonlinear Schrödinger euqtion with quartic potential on ℝ, J. Statist. Phys. 91, 1997, pp. 439-458.
48. Nawa, H. and Tsutsumi, M. On blowup for the pseudo-conformally invariant nonlinear Schrödinger equation, Funkcial. Ekvac. 32, 1989, pp. 417-428.
49. Nawa, H. and Tsutsumi, M. On blowup for the pseudo-conformally invariant nonlinear Schrödinger equation II, Comm. Pure Appl. Math. 51, 1998, pp. 373-383.
50. Niederer, U., The maximal kinematical invariance group of the free Schrödinger equation, Helv. Phys. Acta. 45, 1972, pp. 802-810.
51. 1-solution for the nonlinear Schrödinger equation, J. Differential Equations 92, 1991, pp. 317-330.
52. 1-solution for the one-dimensional nonlinear Schrödinger equation with critical power nonlinearity, Proc. Amer. Math. Soc. 111, 1991, pp. 487-496.
53. Petrina, D. Ya., Mathematical Foundations of Quantum Statistical Mechanics, Mathematical Physics Studies No. 17, Kluwer, Dordrecht, 1995.
54. Sacks, J., and Uhlenbeck, K., The existence of minimal immersions of 2-spheres, Ann. of Math. (2) 113, 1981, pp. 1-24.
55. Shiryaev, A. N., Probability, 2nd ed., Graduate Texts in Mathematics No. 95, Springer-Verlag, New York, 1996.
56. Strauss, W. A., Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55, 1977, pp. 149-162.
57. Struwe, M., A global compactness result for elliptic boundary value problem involving limiting nonlinearities, Math. Z. 187, 1984, pp. 511-517.
58. Tsutsumi, M., Nonexistence of global solutions to the Cauchy problem for the damped nonlinear Schödinger equations, SIAM J. Math. Anal. 15, 1984, pp. 357-366.
59. Tsutsumi, M., Nonexistence and instability of solutions of nonlinear Schödinger equations, unpublished manuscript.
60. 2 concentration of blowup solutions for the nonlinear Schrödinger equation with critical power nonlinearity, Nonlinear Anal. 15, 1990, pp. 719-724.
61. Tsutsumi, Y., Lower estimates of blowup solutions for nonlinear Schrödinger equations, unpublished manuscript.
62. Weinstein, M. I., Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87, 1982/1983, pp. 567-576.
63. Weinstein, M. I., On the structure and formation singularities in solutions to nonlinear dispersive evolution equations, Comm. Partial Differential Equations 11, 1986, pp. 545-565.
64. Weinstein, M. I., The nonlinear Schrödinger equation-singularity formation, stability and dispersion, pp. 213-232 in: The Connection Between Infinite-Dimensional and Finite-Dimensional Dynamical Systems, Contemporary Math. No. 99, Amer. Math. Soc., Providence, R.I., 1989.
65. Yajima, K., Existence of solutions for Schrödinger evolution equations, Comm. Math. Phys. 110, 1987, pp. 415-426.
66. Zakharov, V. E., Collapse of Langmuir waves, Sov. Phys. JETP 35, 1972, pp. 908-914.
67. Zakharov, V. E., and Synakh, V. S., The nature of self-focusing singularity, Sov. Phys. JETP 41, 1976, pp. 441-448.