Changing blow-up time in nonlinear Schrödinger equations
Journées équations aux dérivées partielles (2003), article no. 3, 12 p.

Solutions to nonlinear Schrödinger equations may blow up in finite time. We study the influence of the introduction of a potential on this phenomenon. For a linear potential (Stark effect), the blow-up time remains unchanged, but the location of the collapse is altered. The main part of our study concerns isotropic quadratic potentials. We show that the usual (confining) harmonic potential may anticipate the blow-up time, and always does when the power of the nonlinearity is L 2 -critical. On the other hand, introducing a “repulsive” harmonic potential prevents finite time blow-up, provided that this potential is sufficiently “strong”. For the L 2 -critical nonlinearity, this mechanism is explicit : according to the strength of the potential, blow-up is first delayed, then prevented.

     author = {Carles, R\'emi},
     title = {Changing blow-up time in nonlinear Schr\"odinger equations},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {3},
     publisher = {Universit\'e de Nantes},
     year = {2003},
     doi = {10.5802/jedp.617},
     zbl = {02079438},
     mrnumber = {2050589},
     language = {en},
     url = {}
Carles, Rémi. Changing blow-up time in nonlinear Schrödinger equations. Journées équations aux dérivées partielles (2003), article  no. 3, 12 p. doi : 10.5802/jedp.617.

[1] J. Bourgain and W. Wang, Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), no. 1-2, 197-215 (1998). | Numdam | MR 1655515 | Zbl 1043.35137

[2] C. C. Bradley, C. A. Sackett, and R. G. Hulet, Bose-Einstein condensation of Lithium: Observation of limited condensate number, Phys. Rev. Lett. 78 (1997), 985-989.

[3] C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, Evidence of Bose-Einstein condensation in an atomic gas with attractive interactions, Phys. Rev. Lett. 75 (1995), 1687-1690.

[4] R. Carles, Remarks on nonlinear Schrödinger equations with harmonic potential, Ann. H. Poincaré 3 (2002), no. 4, 757-772. | MR 1933369 | Zbl 1021.81013

[5] R. Carles, Critical nonlinear Schrödinger equations with and without harmonic potential, Math. Mod. Meth. Appl. Sci. (M3AS) 12 (2002), no. 10, 1513-1523. | MR 1933935 | Zbl 1029.35208

[6] R. Carles, Semi-classical Schrödinger equations with harmonic potential and nonlinear perturbation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), no. 3, 501-542. | Numdam | MR 1972872 | Zbl 1031.35119

[7] R. Carles, Nonlinear Schrödinger equations with repulsive harmonic potential and applications, SIAM J. Math. Anal., to appear. | MR 2049023 | Zbl 1054.35090

[8] R. Carles, C. Fermanian, and I. Gallagher, On the role of quadratic oscillations in nonlinear Schrödinger equations, J. Funct. Anal., to appear. | MR 2003356 | Zbl 02005497

[9] R. Carles and Y. Nakamura, Nonlinear Schrödinger equations with Stark potential, Hokkaido Math. J., to appear. | MR 2049023 | Zbl 1069.35083

[10] T. Cazenave, An introduction to nonlinear Schrödinger equations, Text. Met. Mat., vol. 26, Univ. Fed. Rio de Jan., 1993.

[11] H. L. Cycon, R. G. Froese, W. Kirsch, and B. Simon, Schrödinger operators with application to quantum mechanics and global geometry, study ed., Texts and Monographs in Physics, Springer-Verlag, Berlin, 1987. | MR 883643 | Zbl 0619.47005

[12] T. Cazenave and F. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger equation, Comm. Math. Phys. 147 (1992), 75-100. | MR 1171761 | Zbl 0763.35085

[13] A. De Bouard and A. Debussche, Explosion en temps fini pour l'équation de Schrödinger non linéaire stochastique, Séminaire X-EDP, 2002-2003, École Polytech., Exp. No. VII. | Numdam | MR 2030702

[14] A. De Bouard, A. Debussche and L. Di Menza, Theoretical and numerical aspects of stochastic nonlinear Schrödinger equations, Journées équations aux Dérivées Partielles, Plestin-les-Grèves, 2001, Exp. No. III. | Numdam | MR 1843404 | Zbl 1005.35084

[15] R. P. Feynman and A.R. Hibbs, Quantum mechanics and path integrals (International Series in Pure and Applied Physics), Maidenhead, Berksh.: McGraw-Hill Publishing Company, Ltd., 365 p., 1965. | Zbl 0176.54902

[16] R. T. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys. 18 (1977), 1794-1797. | MR 460850 | Zbl 0372.35009

[17] E. B. Kolomeisky, T. J. Newman, J. P. Straley, and X. Qi, Low-dimensional Bose liquids: Beyond the Gross-Pitaevskii approximation, Phys. Rev. Lett. 85 (2000), no. 6, 1146-1149.

[18] F. Merle, Blow-up phenomena for critical nonlinear Schrödinger and Zakharov equations, Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998), no. Extra Vol. III, 1998, pp. 57-66. | MR 1648140 | Zbl 0896.35123

[19] F. Merle and P. Raphaël, Blow up Dynamic and Upper Bound on the Blow up Rate fir critical nonlinear Schrödinger Equation, Journées équations aux Dérivées Partielles, Forges-les-Eaux, 2002, Exp. No. XII. | Numdam | MR 1968208

[20] F. Merle and P. Raphaël, Sharp upper bound on blow up rate for critical non linear Schrödinger equation, Geom. Funct. Anal., to appear. | MR 1995801 | Zbl 01973268

[21] G. Perelman, On the formation of singularities in solutions of the critical nonlinear Schrödinger equation, Ann. H. Poincaré 2 (2001), no. 4, 605-673. | MR 1852922 | Zbl 1007.35087

[22] W. Thirring, A course in mathematical physics. Vol. 3, Springer-Verlag, New York, 1981, Quantum mechanics of atoms and molecules, Translated from the German by Evans M. Harrell, Lecture Notes in Physics, 141. | MR 625662 | Zbl 0462.46046

[23] T. Tsurumi, H. Morise, and M. Wadati, Stability of Bose-Einstein condensates confined in traps, Internat. J. Modern Phys. B 14 (2000), no. 7, 655-719.