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Soliton solutions for quasilinear Schrödinger equations, I


Authors: Jiaquan Liu and Zhi-Qiang Wang
Journal: Proc. Amer. Math. Soc. 131 (2003), 441-448
MSC (2000): Primary 35J10, 35J20, 35J25
DOI: https://doi.org/10.1090/S0002-9939-02-06783-7
Published electronically: September 17, 2002
MathSciNet review: 1933335
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Abstract: For a class of quasilinear Schrödinger equations we establish the existence of ground states of soliton type solutions by a minimization argument.


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  • 1. Bartsch, T., Wang, Z.-Q.: Existence and multiplicity results for some superlinear elliptic problems on ${\mathbf{R}}^n.$ Comm. Partial Differential Equations 20, 1725-1741 (1995). MR 96f:35050
  • 2. Bass, F. G., Nasanov, N. N.: Nonlinear electromagnetic spin waves. Physics Reports 189, 165-223 (1990).
  • 3. Berestycki, H., Lions, P. L.: Nonlinear scalar field equations, I: Existence of a ground state. Arch. Rational Mech. Anal. 82, 313-346 (1983). MR 84h:35054a
  • 4. Borovskii, A. V., Galkin, A. L.: Dynamical modulation of an ultrashort high-intensity laser pulse in matter. JETP 77, 562-573 (1993).
  • 5. De Bouard, A., Hayashi, N., Saut, J.-C.: Global existence of small solutions to a relativistic nonlinear Schrödinger equation. Comm. Math. Phys. 189, 73-105 (1997). MR 98k:35174
  • 6. Brandi, H. S., Manus, C., Mainfray, G., Lehner, T., Bonnaud, G.: Relativistic and ponderomotive self-focusing of a laser beam in a radially inhomogeneous plasma. Phys. Fluids B 5, 3539-3550 (1993).
  • 7. Browder, F. E.: Variational methods for nonlinear elliptic eigenvalue problems. Bull. Amer. Math. Soc. 71, 176-183 (1965). MR 31:3707
  • 8. Chen, X. L., Sudan, R. N.: Necessary and sufficient conditions for self-focusing of short ultraintense laser pulse. Phys. Review Letters 70, 2082-2085 (1993)
  • 9. Floer, A., Weinstein, A.: Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal. 69, 397-408 (1986). MR 88d:35169
  • 10. Hasse, R. W.: A general method for the solution of nonlinear soliton and kink Schrödinger equations. Z. Physik B 37, 83-87 (1980). MR 80m:81017
  • 11. Kondrat'ev, V., Shubin, M.:, Discreteness of spectrum for the Schrödinger operator on manifolds of bounded geometry. Operator theory: Advances and Applications, 110 (1999), 185-226. MR 2001c:58030
  • 12. Kosevich, A. M., Ivanov, B. A., Kovalev, A. S.: Magnetic solitons. Physics Reports 194, 117-238 (1990)
  • 13. Kurihura, S.: Large-amplitude quasi-solitons in superfluid films. J. Phys. Soc. Japan 50, 3262-3267 (1981).
  • 14. Laedke, E. W., Spatschek, K. H., Stenflo, L.: Evolution theorem for a class of perturbed envelope soliton solutions. J. Math. Phys. 24, 2764-2769 (1983). MR 85e:76012
  • 15. Lange, H., Toomire, B., Zweifel, P. F.: Time-dependent dissipation in nonlinear Schrödinger systems. J. Math. Phys. 36, 1274-1283 (1995). MR 95k:35193
  • 16. Lions, P.-L.: Concentration compactness principle in the calculus of variations. The limit case. Part 1, Rev. Mat. Ibero., 1, 145-201 (1985). MR 87c:49007
  • 17. Litvak, A. G., Sergeev, A. M.: One dimensional collapse of plasma waves. JETP Letters 27, (1978) 517-520.
  • 18. Makhankov, V. G., Fedyanin, V. K.: Non-linear effects in quasi-one-dimensional models of condensed matter theory. Physics Reports 104, 1-86 (1984). MR 85b:82003
  • 19. Nakamura, A.: Damping and modification of exciton solitary waves. J. Phys. Soc. Japan 42, 1824-1835 (1977).
  • 20. Poppenberg, M., Schmitt, K., Wang, Z.-Q.: On the existence of soliton solutions to quasilinear Schrödinger equations. Calculus of Variations and PDEs 14, 329-344 (2002).
  • 21. Porkolab, M., Goldman, M. V.: Upper hybrid solitons and oscillating two-stream instabilities. Physics of Fluids 19, 872-881 (1976). MR 54:14658
  • 22. Quispel, G. R. W., Capel, H. W.: Equation of motion for the Heisenberg spin chain. Physica 110 A, 41-80 (1982). MR 83g:82071
  • 23. Ritchie, B.: Relativistic self-focusing and channel formation in laser-plasma interactions. Phys. Rev. E 50, 687-689 (1994).
  • 24. Strauss, W. A.:Existence of solitary waves in higher dimensions. Comm. Math. Phys., 55, 149-162 (1977). MR 56:12616
  • 25. Takeno, S., Homma, S.: Classical planar Heisenberg ferromagnet, complex scalar fields and nonlinear excitations. Progr. Theoret. Physics 65, 172-189 (1981).
  • 26. Willem, M.: Minimax Theorems. Birkhäuser, Boston, (1996). MR 97h:58037

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Additional Information

Jiaquan Liu
Affiliation: Department of Mathematics, Peking University, Beijing, 100871, People’s Republic of China
Email: jiaquan@math.pku.edu.cn

Zhi-Qiang Wang
Affiliation: Department of Mathematics and Statistics, Utah State University, Logan, Utah 84322
Email: wang@math.usu.edu

DOI: https://doi.org/10.1090/S0002-9939-02-06783-7
Received by editor(s): September 4, 2001
Published electronically: September 17, 2002
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2002 American Mathematical Society

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