Variational method: How it can generate false instabilities

Journal of Mathematical Physics

Volumn 37, Issue 7, 1996, Pages 3442-3462

  Kaup, D J ^a   Lakoba, T I ^a  

^a CLARKSON UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0030538778     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.531574     Document Type: Article

References (14)

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2. D. J. Kaup, B. A. Malomed, and R. S. Tasgal, "Internal dynamics of a vector soliton in nonlinear optical fibers," Phys. Rev. E 48, 3049-3053 (1993).
3. B. A. Malomed and R. S. Tasgal, "Vibrational modes of a gap soliton in a nonlinear optical medium," Phys. Rev. E 49, 5787-5796 (1994).
4. E. W. Laedke and K. H. Spatschek, "Variational formulation for the evolution of a deep-water wavetrains in 2 dimensions," Phys. Fluids 28, 2315-2316 (1985); "Thresholds and growth times of some nonlinear field equations," Physica D 5, 227-242 (1982).
5. E. W. Laedke and K. H. Spatschek, "Variational formulation for the evolution of a deep-water wavetrains in 2 dimensions," Phys. Fluids 28, 2315-2316 (1985); "Thresholds and growth times of some nonlinear field equations," Physica D 5, 227-242 (1982).
6. E. W. Laedke and K. H. Spatschek, "Nonlinear stability of envelope solitons," Phys. Rev. Lett. 41, 1798-1801 (1978).
7. I. V. Barashenkov, "Stability properties of solutions to nonlinear models possessing a sign-undefinite metric," Acta Phys. Austriaca 55, 155-165 (1983).
8. D. J. Kaup, "Perturbation theory for solitons in optical fibers," Phys. Rev. A 42, 5689-5694 (1990).
9. F. B. Hildebrand, Methods of Applied Mathematics (Dover, New York, 1992), Sec. 1.25.
10. F. R. Gantmakher, The Theory of Matrices (Chelsea, New York, 1960), Vol. 1.
11. 1j is not at all restrictive since any reasonable ansatz must have this property (otherwise, roughly speaking, we will have one variation to which there correspond two variational parameters). Then we could simply delete one variational parameter.
12. E. A. Kuznetzov and A. V. Mikhailov, "The complete integrability of the two-dimensional classical Thirring model," Theor. Math. Phys. 30, 193-200 (1977) (translated from Teor. Mat. Fiz.).
13. D. Kaup and T. I. Lakoba, "The squared eigefunctions of the Massive Thirring Model in laboratory coordinates," J. Math. Phys. 37, 308-323 (1996).
14. J. M. Soto-Crespo, and N. Akhmediev, "Stability of the soliton states in a nonlinear optical coupler," Phys. Rev. E 48, 4710-4715 (1993).