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Proceedings of the American Mathematical Society

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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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Soliton solutions for quasilinear Schrödinger equations, I
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by Jiaquan Liu and Zhi-Qiang Wang PDF
Proc. Amer. Math. Soc. 131 (2003), 441-448 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
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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
  • 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