Abstract
We apply a linear perturbation analysis to investigate the relationship between soliton oscillations and the integrability of nonlinear PDEs in bi-dimensional spacetime. For this purpose, we consider a localized solution of the nonlinear differential equation, and study small amplitude fluctuations around it. The linearized equation is a Schrödinger-like, eigenvalue problem. By considering several nonlinear PDEs, which are known to have soliton and solitary wave solutions, we find that in systems which are integrable, this eigenvalue equation has one and only one bound state with zero frequency. Non-integrable equations—in contrast—show extra bound states. The time evolution of the oscillations are also calculated, using a numerical program to integrate the time-dependent equation. The behavior of the modes are studied, using the Fourier transform of the evolving solutions.
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References
Berry, M. V. (1978). AIP Proceedings 46, 16.
Das, A. (1989). Integrable Models, World Scientific.
Drazin, P. G. and Johnson, R. S. (1989). Solitons: An Introduction, Cambridge University Press, Cambridge.
Hasegawa, A. and Kodama, Y. (1995). Solitons in Optical Communications, Oxford University Press.
Ince, E. L. (1927). Ordinary Differential Equations, Longman, London.
Khare, A., Habib, S., and Saxena, A. (1997). Physical Review Letters 79(20), 3797.
Lamb, G. L., Jr. (1980). Elements of Soliton Theory, John Wiley and Sons, New York, U.S.A.
Lax, P. D. (1968). Communications on Pure and Applied Mathematics 21, 467.
Pelinovsky, D. E., Kivshar, Y. S., and Afanasjev, V. V. (1998). Physica D 116, 121.
Rajaraman, R. (1982). Solitons and Instantons, North Holland, Elsevier, Amsterdam.
Ramani, A., Grammaticos, B., and Bountis, T. (1989). Physics Reports 180(3), 159–245.
Riazi, N. (1993). International Journal of Theoretical Physics 32(11), 2155.
Riazi, N. and Gharaati, A. R. (1998). International Journal of Theoretical Physics 37(3), 1081.
Turitsyn, S. K. (1993). Physical Review E47, R796.
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Gharaati, A.R., Riazi, N. & Mohebbi, a.F. Internal Modes of Relativistic Solitons. Int J Theor Phys 45, 53–63 (2006). https://doi.org/10.1007/s10773-005-9009-8
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DOI: https://doi.org/10.1007/s10773-005-9009-8