Sharp Conditions of Global Existence for the Quasilinear Schr\"{o}dinger Equation

J. Zhang ; J. Shu

Funkcionalʹnyj analiz i ego priloženiâ, Tome 42 (2008) no. 2, pp. 68-74.

Résumé

The paper discusses a class of quasilinear Schrödinger equations describing the upper-hybrid oscillation propagation. By establishing a cross-constrained variational problem and a so-called invariant flow, we obtain a sharp condition for blow-up and global existence of solutions of the Cauchy problem.

Détail

Keywords: quasilinear Schrödinger equation, blow-up, sharp condition.
Mots-clés : global existence

Bibliographie

[1] S. Kurihura, “Large-amplitude quasi-solitons in superfluid films”, J. Phys. Soc. Japan, 50:10 (1981), 3262–3267 | DOI | MR

[2] H. Lange, M. Poppenberg, H. Teismann, “Nash–Moser methods for the solution of quasi-linear Schrödinger equations”, Comm. Partial Differential Equations, 24:7–8 (1999), 1399–1418 | DOI | MR | Zbl

[3] E. W. Laedke, K. H. Spatschek, L. Stenflo, “Evolution theorem for a class of perturbed envelope soliton solutions”, J. Math. Phys., 24:12 (1983), 2764–2769 | DOI | MR | Zbl

[4] A. V. Borovskii, A. L. Galkin, “Dinamicheskaya modulyatsiya ultrakorotkogo intensivnogo lazernogo impulsa v veschestve”, ZhETF, 104:10 (1993), 3311–3333

[5] H. S. Brandi, C. Manus, G. Mainfray, T. Lehner, G. Bonnaud, “Relativistic and ponderomotive self-focusing of a laser beam in a radially inhomogeneous plasma”, Phys. Fluids B, 5 (1993), 3539–3550 | DOI

[6] X. L. Chen, R. N. Sudan, “Necessary and sufficient conditions for self-focusing of short ultraintense laser pulse in underdence plasma”, Phys. Rev. Lett., 70:14 (1993), 2082–2085 | DOI

[7] B. Ritchie, “Relativistic self-focusing and channel formation in laser-plasma interactions”, Phys. Rev. E, 50:2 (1994), 687–689 | DOI

[8] A. De Bouard, N. Hayashi, J. C. Saut, “Global existence of small solutions to a relativistic nonlinear Schrödinger equation”, Comm. Math. Phys., 189:1 (1997), 73–105 | DOI | MR | Zbl

[9] A. G. Litvak, A. M. Sergeev, “Ob odnomernom kollapse plazmennykh voln”, Pisma v ZhETF, 27:10 (1978), 549–553

[10] A. Nakamura, “Damping and modification of exciton solitary waves”, J. Phys. Soc. Japan, 42:6 (1977), 1824–1835 | DOI

[11] M. Porkolab, M. V. Goddman, “Upper-hybrid solitons and oscillating-two-stream instabilities”, Phys. Fluids, 19:6 (1976), 872–881 | DOI | MR

[12] F. G. Bass, N. N. Nasanov, “Nonlinear electromagnetic spin waves”, Phys. Rep., 189:4 (1990), 165–223 | DOI

[13] A. M. Kosevich, B. A. Ivanov, A. S. Kovalev, “Magnetic solitons”, Phys. Rep., 194:3–4 (1990), 117–238 | DOI

[14] H. Lange, B. Toomire, P. E. Zweifel, “Time-dependent dissipation in nonlinear Schrödinger systems”, J. Math. Phys., 36:3 (1995), 1274–1283 | DOI | MR | Zbl

[15] G. R. W. Quispel, H. W. Capel, “Equation of motion for the Heisenberg spin chain”, Physica A, 110:1–2 (1982), 41–80 | DOI | MR

[16] S. Takeno, S. Homma, “Classical planar Heisenberg ferromagnet, complex scalar fields and nonlinear excitations”, Progr. Theoret. Phys., 65:1 (1981), 172–189 | DOI | MR

[17] R. W. Hasse, “A general method for the solution of nonlinear soliton and kink Schrödinger equations”, Z. Phys. B, 37:1 (1980), 83–87 | DOI | MR

[18] V. G. Makhankov, V. K. Fedyanin, “Non-linear effects in quasi-one-dimensional models of condensed matter theory”, Phys. Rep., 104:1 (1984), 1–86 | DOI | MR

[19] M. Poppenberg, “Smooth solutions for a class of fully nonlinear Schrödinger type equations”, Nonlinear Anal., 45:6 (2001), 723–741 | DOI | MR | Zbl

[20] M. Poppenberg, “On the local well posedness of quasi-linear Schrödinger equations in arbitrary space dimension”, J. Differential Equations, 172:1 (2001), 83–115 | DOI | MR | Zbl

[21] M. Poppenberg, “An inverse function theorem in Sobolev spaces and applications to quasi-linear Schrödinger equations”, J. Math. Anal. Appl., 258:1 (2001), 146–170 | DOI | MR | Zbl

[22] M. Poppenberg, K. Schmitt, Z. Q. Wang, “On the existence of soliton solutions to quasi-linear Schrödinger equations”, Calc. Var. Partial Differential Equations, 14:3 (2002), 329–344 | DOI | MR | Zbl

[23] J. Q. Liu, Y. Q. Wang, Z. Q. Wang, “Soliton solutions for quasi-linear Schrödinger equations II”, J. Differential Equations, 187:2 (2003), 473–493 | DOI | MR | Zbl

[24] J. J. García-Ripoll, V. V. Konotop, B. Malomed, V. M. Pérez-García, “A quasi-local Gross–Pitaevskii equation for attractive Bose–Einstein condensates”, Math. Comput. Simulation, 62:1–2 (2003), 21–30 | DOI | MR | Zbl

[25] J. Zhang, “The blowup properties of the initial-boundary problem for second order derivative nonlinear Schrödinger equations”, Acta Math. Sci., 14 (1994), 89–94

[26] H. Berestycki, T. Cazenave, “Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéarires”, C. R. Acad. Sci. Paris, Seire I, 293 (1981), 489–492 | MR | Zbl

[27] M. I. Weinstein, “Nonlinear Schrödinger equations and sharp interpolations estimates”, Comm. Math. Phys., 87 (1983), 567–576 | DOI | MR | Zbl

[28] J. Zhang, “Sharp conditions of global existence for nonlinear Schrödinger and Klein–Gordon equations”, Nonlinear Anal., 48:2 (2002), 191–207 | DOI | MR

[29] J. Zhang, “Sharp threshold of blowup and global existence in nonlinear Schrödinger equations under a harmonic potential”, Comm. Partial Differential Equations, 30:10–12 (2005), 1429–1443 | DOI | MR | Zbl

[30] L. E. Payne, D. H. Sattinger, “Saddle points and instability of nonlinear hyperbolic equations”, Israel J. Math., 22:3–4 (1975), 273–303 | DOI | MR

[31] H. A. Levine, “Instability and non-existence of global solutions to nonlinear wave equations of the form $Pu_{tt}=-Au+F(u)$”, Trans. Amer. Math. Soc., 192 (1974), 1–21 | DOI | MR | Zbl

[32] T. Cazenave, An introduction to nonlinear Schrödinger equations, Textos de Metodos Mathematicos, 26, Rio de Janeiro, 1993

[33] Y. Tsutsumi, J. Zhang, “Instability of optical solitons for two-wave interaction model in cubic nonlinear media”, Adv. Math. Sci. Appl., 8:2 (1998), 691–713 | MR | Zbl