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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Hydrodynamic approach to constructing solutions of the nonlinear Schrödinger equation in the critical case
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by O. S. Rozanova PDF
Proc. Amer. Math. Soc. 133 (2005), 2347-2358 Request permission

Abstract:

Proceeding from the hydrodynamic approach, we construct exact solutions to the nonlinear Schrödinger equation with special properties. The solutions describe collapse, in finite time, and scattering, over infinite time, of wave packets. They generalize known blow-up solutions based on the “ground state”.
References
  • J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Schrödinger equation revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), no. 4, 309–327 (English, with French summary). MR 801582
  • J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. II. Scattering theory, general case, J. Functional Analysis 32 (1979), no. 1, 33–71. MR 533219, DOI 10.1016/0022-1236(79)90077-6
  • Tosio Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor. 46 (1987), no. 1, 113–129 (English, with French summary). MR 877998
  • Thierry Cazenave and Fred B. Weissler, Some remarks on the nonlinear Schrödinger equation in the critical case, Nonlinear semigroups, partial differential equations and attractors (Washington, DC, 1987) Lecture Notes in Math., vol. 1394, Springer, Berlin, 1989, pp. 18–29. MR 1021011, DOI 10.1007/BFb0086749
  • Björn Birnir, Carlos E. Kenig, Gustavo Ponce, Nils Svanstedt, and Luis Vega, On the ill-posedness of the IVP for the generalized Korteweg-de Vries and nonlinear Schrödinger equations, J. London Math. Soc. (2) 53 (1996), no. 3, 551–559. MR 1396718, DOI 10.1112/jlms/53.3.551
  • Michael I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 (1982/83), no. 4, 567–576. MR 691044
  • R. T. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys. 18 (1977), no. 9, 1794–1797. MR 460850, DOI 10.1063/1.523491
  • Frank Merle, Blow-up phenomena for critical nonlinear Schrödinger and Zakharov equations, Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998), 1998, pp. 57–66. MR 1648140
  • V.I.Talanov, Self-focusing of wave beams in nonlinear media. JETP Lett. 2(1965), 138.
  • Takayoshi Ogawa and Yoshio Tsutsumi, Blow-up of $H^1$ solution for the nonlinear Schrödinger equation, J. Differential Equations 92 (1991), no. 2, 317–330. MR 1120908, DOI 10.1016/0022-0396(91)90052-B
  • Hayato Nawa, Asymptotic and limiting profiles of blowup solutions of the nonlinear Schrödinger equation with critical power, Comm. Pure Appl. Math. 52 (1999), no. 2, 193–270. MR 1653454, DOI 10.1002/(SICI)1097-0312(199902)52:2<193::AID-CPA2>3.0.CO;2-3
  • E.Madelung, Quantentheorie in hydrodynamischer form, Z.Phys. 40(1926), 322.
  • Denis Serre, Solutions classiques globales des équations d’Euler pour un fluide parfait compressible, Ann. Inst. Fourier (Grenoble) 47 (1997), no. 1, 139–153 (French, with English and French summaries). MR 1437182
  • Olga S. Rozanova, Classes of smooth solutions to multidimensional balance laws of gas dynamic type on Riemannian manifolds, Trends in mathematical physics research, Nova Sci. Publ., Hauppauge, NY, 2004, pp. 155–204. MR 2078813
  • Olga S. Rozanova, Solutions with linear profile of velocity to the Euler equations in several dimensions, Hyperbolic problems: theory, numerics, applications, Springer, Berlin, 2003, pp. 861–870. MR 2053233
  • O. S. Rozanova, Application of integral functionals to the study of the properties of solutions to the Euler equations on Riemannian manifolds, J. Math. Sci. (N.Y.) 117 (2003), no. 5, 4551–4584. Partial differential equations. MR 2027444, DOI 10.1023/A:1025162104780
  • Yoshio Tsutsumi, Rate of $L^2$ concentration of blow-up solutions for the nonlinear Schrödinger equation with critical power, Nonlinear Anal. 15 (1990), no. 8, 719–724. MR 1074950, DOI 10.1016/0362-546X(90)90088-X
  • H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal. 82 (1983), no. 4, 313–345. MR 695535, DOI 10.1007/BF00250555
  • Michael I. Weinstein, On the structure and formation of singularities in solutions to nonlinear dispersive evolution equations, Comm. Partial Differential Equations 11 (1986), no. 5, 545–565. MR 829596, DOI 10.1080/03605308608820435
  • F. Merle, Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power, Duke Math. J. 69 (1993), no. 2, 427–454. MR 1203233, DOI 10.1215/S0012-7094-93-06919-0
  • Frank Merle, On uniqueness and continuation properties after blow-up time of self-similar solutions of nonlinear Schrödinger equation with critical exponent and critical mass, Comm. Pure Appl. Math. 45 (1992), no. 2, 203–254. MR 1139066, DOI 10.1002/cpa.3160450204
  • Wei-Ming Ni and Roger D. Nussbaum, Uniqueness and nonuniqueness for positive radial solutions of $\Delta u+f(u,r)=0$, Comm. Pure Appl. Math. 38 (1985), no. 1, 67–108. MR 768105, DOI 10.1002/cpa.3160380105
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Additional Information
  • O. S. Rozanova
  • Affiliation: Department of Differential Equations, Mathematics and Mechanics Faculty, Moscow State University, GSP-2 Vorobiovy Gory, Moscow 119992, Russia
  • Email: rozanova@mech.math.msu.su
  • Received by editor(s): December 10, 2003
  • Published electronically: March 22, 2005
  • Communicated by: Mark J. Ablowitz
  • © Copyright 2005 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 133 (2005), 2347-2358
  • MSC (2000): Primary 35Q55; Secondary 35K55
  • DOI: https://doi.org/10.1090/S0002-9939-05-07920-7
  • MathSciNet review: 2138877