Skip to Main Content

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 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Soliton solutions for quasilinear Schrödinger equations, I
HTML articles powered by AMS MathViewer

by Jiaquan Liu and Zhi-Qiang Wang
Proc. Amer. Math. Soc. 131 (2003), 441-448
DOI: https://doi.org/10.1090/S0002-9939-02-06783-7
Published electronically: September 17, 2002
PDF | Request permission

Abstract:

For a class of quasilinear Schrödinger equations we establish the existence of ground states of soliton type solutions by a minimization argument.
References
  • Thomas Bartsch and Zhi Qiang Wang, Existence and multiplicity results for some superlinear elliptic problems on $\textbf {R}^N$, Comm. Partial Differential Equations 20 (1995), no. 9-10, 1725–1741. MR 1349229, DOI 10.1080/03605309508821149
  • Bass, F. G., Nasanov, N. N.: Nonlinear electromagnetic spin waves. Physics Reports 189, 165-223 (1990).
  • 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
  • Borovskii, A. V., Galkin, A. L.: Dynamical modulation of an ultrashort high-intensity laser pulse in matter. JETP 77, 562-573 (1993).
  • Anne de Bouard, Nakao Hayashi, and Jean-Claude Saut, Global existence of small solutions to a relativistic nonlinear Schrödinger equation, Comm. Math. Phys. 189 (1997), no. 1, 73–105 (English, with English and French summaries). MR 1478531, DOI 10.1007/s002200050191
  • 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).
  • Felix E. Browder, Variational methods for nonlinear elliptic eigenvalue problems, Bull. Amer. Math. Soc. 71 (1965), 176–183. MR 179459, DOI 10.1090/S0002-9904-1965-11275-7
  • 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)
  • Andreas Floer and Alan Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal. 69 (1986), no. 3, 397–408. MR 867665, DOI 10.1016/0022-1236(86)90096-0
  • Rainer W. Hasse, A general method for the solution of nonlinear soliton and kink Schrödinger equations, Z. Phys. B 37 (1980), no. 1, 83–87. MR 563644, DOI 10.1007/BF01325508
  • Vladimir Kondrat′ev and Mikhail Shubin, Discreteness of spectrum for the Schrödinger operators on manifolds of bounded geometry, The Maz′ya anniversary collection, Vol. 2 (Rostock, 1998) Oper. Theory Adv. Appl., vol. 110, Birkhäuser, Basel, 1999, pp. 185–226. MR 1747895
  • Kosevich, A. M., Ivanov, B. A., Kovalev, A. S.: Magnetic
  • osevich, A. M., Ivanov, B. A., Kovalev, A. S.: Magnetic solitons. Physics Reports 194, 117-238 (1990) Kuri Kurihura, S.: Large-amplitude quasi-solitons in superfluid films. J. Phys. Soc. Japan 50, 3262-3267 (1981).
  • E. W. Laedke, K. H. Spatschek, and L. Stenflo, Evolution theorem for a class of perturbed envelope soliton solutions, J. Math. Phys. 24 (1983), no. 12, 2764–2769. MR 727767, DOI 10.1063/1.525675
  • H. Lange, B. Toomire, and P. F. Zweifel, Time-dependent dissipation in nonlinear Schrödinger systems, J. Math. Phys. 36 (1995), no. 3, 1274–1283. MR 1317440, DOI 10.1063/1.531120
  • P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana 1 (1985), no. 1, 145–201. MR 834360, DOI 10.4171/RMI/6
  • Litvak, A. G., Sergeev, A. M.: One dimensional collapse of plasma waves. JETP Letters 27, (1978) 517-520.
  • V. G. Makhan′kov and V. K. Fedyanin, Nonlinear effects in quasi-one-dimensional models and condensed matter theory, Phys. Rep. 104 (1984), no. 1, 1–86. MR 740342, DOI 10.1016/0370-1573(84)90106-6
  • Nakamura, A.: Damping and modification of exciton solitary waves. J. Phys. Soc. Japan 42, 1824-1835 (1977).
  • 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).
  • Miklos Porkolab and Martin V. Goldman, Upper-hybrid solitons and oscillating-two-stream instabilities, Phys. Fluids 19 (1976), no. 6, 872–881. MR 426721, DOI 10.1063/1.861553
  • G. R. W. Quispel and H. W. Capel, Equation of motion for the Heisenberg spin chain, Phys. A 110 (1982), no. 1-2, 41–80. MR 647411, DOI 10.1016/0378-4371(82)90104-2
  • Ritchie, B.: Relativistic self-focusing and channel formation in laser-plasma interactions. Phys. Rev. E 50, 687-689 (1994).
  • Walter A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), no. 2, 149–162. MR 454365
  • Takeno, S., Homma, S.: Classical planar Heisenberg ferromagnet, complex scalar fields and nonlinear excitations. Progr. Theoret. Physics 65, 172-189 (1981).
  • Michel Willem, Minimax theorems, Progress in Nonlinear Differential Equations and their Applications, vol. 24, Birkhäuser Boston, Inc., Boston, MA, 1996. MR 1400007, DOI 10.1007/978-1-4612-4146-1
Similar Articles
  • Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 35J10, 35J20, 35J25
  • Retrieve articles in all journals with MSC (2000): 35J10, 35J20, 35J25
Bibliographic 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
  • MR Author ID: 239651
  • Email: wang@math.usu.edu
  • Received by editor(s): September 4, 2001
  • Published electronically: September 17, 2002
  • Communicated by: David S. Tartakoff
  • © Copyright 2002 American Mathematical Society
  • 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
  • MathSciNet review: 1933335