Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-05T19:02:01.749Z Has data issue: false hasContentIssue false
  • English
  • Français

The structure of solutions to the pseudo-conformally invariant nonlinear Schrödinger equation

Published online by Cambridge University Press:  14 November 2011

Thierry Cazenave  and
Fred B. Weissler
Thierry Cazenave
Affiliation:
Analyse Numérique, Université Pierre et Marie Curie, 4, Place Jussieu, 75252 Paris Cedex 05, France
Fred B. Weissler
Affiliation:
Centre de Mathématiques, Ecole Normale Supérieure de Cachan, 61, Avenue du Président Wilson, 94235 Cachan Cedex, France
Get access Rights & Permissions [Opens in a new window]

Synopsis

We study solutions in ℝn of the nonlinear Schrödinger equation iut + Δu = λ |u|γu, where γ is the fixed power 4/n. For this particular power, these solutions satisfy the “pseudo-conformal” conservation law, and the set of solutions is invariant under a related transformation. This transformation gives a correspondence between global and non-global solutions (if λ < 0), and therefore allows us to deduce properties of global solutions from properties of non-global solutions, and vice versa. In particular, we show that a global solution is stable if and only if it decays at the same rate as a solution to the linear problem (with λ = 0). Also, we obtain an explicit formula for the inverse of the wave operator; and we give a sufficient condition (if λ < 0) that the blow up time of a non-global solution is a continuous function on the set of initial values with (for example) negative energy.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 117 , Issue 3-4 , 1991 , pp. 251 - 273
Copyright
Copyright © Royal Society of Edinburgh 1991

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

1Atkinson, F. V. and Peletier, L. A.. Ground states of −Δu = f(u) and the Emden–Fowler equation, Arch. Rational. Mech. Anal. 93 (1986), 103–127.CrossRefGoogle Scholar
2Berestycki, H. and Lions, P.-L.. Nonlinear scalar field equations I, Existence of a ground state; II, Existence of infinitely many solutions. Arch. Rational. Mech. Anal. 82 (1983), 313–375.CrossRefGoogle Scholar
3Berestycki, H., Lions, P.-L. and Peletier, L. A.. An O.D.E. approach to the existence of positive solutions of semilinear problems in ℝn. Indiana Univ. Math. J. 30 (1981), 147–157.CrossRefGoogle Scholar
4Cazenave, T. and Haraux, A.. Introduction aux Problémes d'Evolution Semi-linéaires, Mathématiques et Applications n 1, (Paris: Ellipses, 1990).Google Scholar
5Cazenave, T. and Weissler, F. B.. Some remarks on the nonlinear Schrödinger equation in the critical case. In Nonlinear Semigroups, Partial Differential Equations, and Attractors: Proceedings of a Symposium held in Washington, D.C., August 3–7, 1987, eds Gill, T. L. and Zachary, W. W., pp. 18–29 Lecture Notes in Mathematics 1394 (New York: Springer, 1989).CrossRefGoogle Scholar
6Cazenave, T. and Weissler, F. B.. The Cauchy problem for the critical nonlinear Schrödinger equation in H s. Nonlinear Anal. 14 (1990), 807–836.CrossRefGoogle Scholar
7Cazenave, T. and Weissler, F. B.. Rapidly decaying solutions of the nonlinear Schrödinger equation (preprint).Google Scholar
8Ginibre, J. and Velo, G.. On a class of nonlinear Schrödinger equations I, II. J. Func. Anal. 32 (1979), 1–71.Google Scholar
9Ginibre, J. and Velo, G., Sur une équation de Schrödinger non linéaire avec interaction non locale. In Nonlinear partial differential equations and their applications, Collège de France Seminar, Vol. II, eds Brezis, H. and Lions, J. L., pp. 155–199. (Boston: Pitman, 1982).Google Scholar
10Ginibre, J. and Velo, G.. The global Cauchy problem for the nonlinear Schrödinger equation revisited. Ann Inst. H. Poincaré, Anal. Non Linéaire 2 (1985), 309–327.CrossRefGoogle Scholar
11Glassey, R. T.. On the blowing up of solutions to the Cauchy problem for the nonlinear Schrödinger equation. J. Math. Phys. 18 (1977), 1794–1797.CrossRefGoogle Scholar
12Jones, C. and Küpper, T.. On the infinitely many solutions of a semilinear elliptic equation. SIAM J. Math. Anal. 17 (1986), 803–835.CrossRefGoogle Scholar
13Kato, T.. On nonlinear Schrödinger equations. Ann. Inst. H. Poincaré, Phys. Théor. 46 (1987), 113–129.Google Scholar
14Kwong, M. K.. Uniqueness of positive solutions of Δu − u + u p in ℝp. Arch. Rational. Méch. Anal. 105 (1989), 243–266.CrossRefGoogle Scholar
15Landman, M. J., Papanicolaou, G. C., Sulem, C. and Sulem, P. L., Rate of blow up for solutions of the nonlinear Schroedinger equation at critical dimension. Phys. Rev. A 38 (1988), 3837–3843.Google Scholar
16LeMesurier, B. J., Papanicolaou, G., Sulem, C. and Sulem, P. L., The focusing singularity of the nonlinear Schrödinger equation. In Directions in Partial Differential Equations, eds Crandall, M. G., Rabinowitz, P. H. and Turner, R. E., pp. 159–201 (New York: Academic Press, 1987).CrossRefGoogle Scholar
17LeMesurier, B. J., Papanicolaou, G. C., Sulem, C. and Sulem, P. L.. Local structure of the self-focusing singularity of the nonlinear Schrodinger equation. Phys. D 32 (1988), 210–226.CrossRefGoogle Scholar
18LeMesurier, B. J., Papanicolaou, G., Sulem, C. and Sulem, P. L.. Focusing and Multi-focusing solutions of the nonlinear Schrodinger equation. Phys. D 31 (1988), 78–102.Google Scholar
19McLaughlin, D. W., Papanicolaou, G. C., Sulem, C. and Sulem, P. L.. Focusing singularity of the cubic Schrödinger equation. Phys. Rev. A 34 (1986), 1200–1210.Google Scholar
20McLeod, K. and Serrin, J.. Uniqueness of positive radial solutions of Δu + f(u) = 0 in ℝn. Arch. Rational. Mech. Anal. 99 (1987), 115–145.CrossRefGoogle Scholar
21McLeod, K., Troy, W. C. and Weissler, F. B.. Radial solutions of Δu + f(u) = 0 with prescribed numbers of zeroes. J. Differential Equations 83 (1990), 368–378.CrossRefGoogle Scholar
22Merle, F.. Limit of the solution of the nonlinear Schrödinger equation at the blow-up time. J. Fund. Anal. 84 (1989), 201–214.CrossRefGoogle Scholar
23Merle, F.. Construction of solutions with exactly k blow-up points for the Schrödinger equation with critical nonlinearity, Comm. Math. Phys. 129 (1990), 223–240.CrossRefGoogle Scholar
24Merle, F. and Tsutsumi, Y.. L 2-concentration of blow-up solutions for the nonlinear Schrödinger equation with the critical power nonlinearity J. Differential Equations 84 (1990), 205–214.CrossRefGoogle Scholar
25Nawa, H. and Tsutsumi, M.. On blow-up for the pseudo-conformally invariant nonlinear Schrödinger equation. Funkcial. Ekvac. 32 (1989), 417–428.Google Scholar
26Strauss, W. A.. Existence of solitary waves in higher dimensions. Comm. Math. Phys. 55 (1977), 149–162.CrossRefGoogle Scholar
27Tsutsumi, Y.. Scattering problem for nonlinear Schrödinger equations. Ann. Inst. H. Poincaré 43 (1985), 321–347.Google Scholar
28Tsutsumi, Y.. L 2-solutions for nonlinear Schrödinger equations and nonlinear groups. Funkcial. Ekvac. 30 (1987), 115–125.Google Scholar
29Tsutsumi, Y.. Rate of L 2 concentration of blow-up solutions for the nonlinear Schrödinger equation with the critical power nonlinearity. Nonlinear Anal, (to appear).Google Scholar
30Weinstein, M. I.. Nonlinear Schrödinger equations and sharp interpolation estimates. Comm. Math. Phys. 87 (1983), 567–576.Google Scholar
31Weinstein, M. I.. On the structure and formation of singularities of solutions to nonlinear dispersive equations. Comm. Partial Differential Equations 11 (1986), 545–565.CrossRefGoogle Scholar
32Weinstein, M. I.. Solitary waves of nonlinear dispersive evolution equations with critical power nonlinearities. J. Differential Equations 69 (1987), 192–203.CrossRefGoogle Scholar
33Weinstein, M. I., The nonlinear Schrö dinger equation – singularity formation, stability, and dispersion. In The Connection between Infinite Dimensional and Finite Dimensional Dynamical Systems, Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference held July 19–25, 1987, eds Nicolaenko, B., Foias, C. and Temam, R. Contemporary Mathematics, Vol. 99, pp. 213–232. (Providence: AMS, 1989).Google Scholar