Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 6 (2010), 017, 22 pages      arXiv:1002.1932      doi:10.3842/SIGMA.2010.017
Contribution to the Proceedings of the Eighth International Conference Symmetry in Nonlinear Mathematical Physics

Solitary Waves in Massive Nonlinear SN-Sigma Models

Alberto Alonso Izquierdo a, Miguel Ángel González León a and Marina de la Torre Mayado b
a) Departamento de Matemática Aplicada, Universidad de Salamanca, Spain
b) Departamento de Física Fundamental, Universidad de Salamanca, Spain

Received December 07, 2009; Published online February 09, 2010

Abstract
The solitary waves of massive (1+1)-dimensional nonlinear SN-sigma models are unveiled. It is shown that the solitary waves in these systems are in one-to-one correspondence with the separatrix trajectories in the repulsive N-dimensional Neumann mechanical problem. There are topological (heteroclinic trajectories) and non-topological (homoclinic trajectories) kinks. The stability of some embedded sine-Gordon kinks is discussed by means of the direct estimation of the spectra of the second-order fluctuation operators around them, whereas the instability of other topological and non-topological kinks is established applying the Morse index theorem.

Key words: solitary waves; nonlinear sigma models.

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References

  1. Alonso Izquierdo A., González León M.A., Mateos Guilarte J., Kinks in a nonlinear massive sigma model, Phys. Rev. Lett. 101 (2008), 131602, 4 pages, arXiv:0808.3052.
  2. Alonso Izquierdo A., González León M.A., Mateos Guilarte J., BPS and non-BPS kinks in a massive nonlinear S2-sigma model, Phys. Rev. D 79 (2009), 125003, 16 pages, arXiv:0903.0593.
  3. Alonso Izquierdo A., González León M.A., Mateos Guilarte J., Senosiain M.J., On the semiclassical mass of S2-kinks, J. Phys. A: Math. Theor. 42 (2009), 385403, 18 pages, arXiv:0906.1258.
  4. Gell-Mann M., Lévy M., The axial vector current in beta decay, Nuovo Cimento (10) 16 (1960), 705-726.
  5. Brézin E., Zinn-Justin J., Le Guillou J.C., Renormalization of the nonlinear σ-model in 2+ε dimensions, Phys. Rev. D 14 (1976), 2615-2621.
  6. Woodford S.R., Barashenkov I.V., Stability of the Bloch wall via the Bogomolnyi decomposition in elliptic coordinates, J. Phys. A: Math. Teor. 41 (2008), 185203, 11 pages, arXiv:0803.2299.
  7. Barashenkov I.V., Woodford S.R., Zemlyanaya E.V., Interactions of parametrically driven dark solitons. I. Néel-Néel and Bloch-Bloch interactions, Phys. Rev. E 75 (2007), 026604, 18 pages, nlin.SI/0612059.
    Barashenkov I.V., Woodford S.R., Interactions of parametrically driven dark solitons. II. Néel-Bloch interactions, Phys. Rev. E 75 (2007), 026605, 14 pages, nlin.SI/0701005.
  8. Barashenkov I.V., Woodford S.R., Zemlyanaya E.V., Parametrically driven dark solitons, Phys. Rev. Lett. 90 (2003), 054103, 4 pages, nlin.SI/0212052.
  9. Rajaraman R., Solitons and instantons. An introduction to solitons and instantons in quantum field theory, North-Holland Publishing Co., Amsterdam, 1982.
  10. Alonso Izquierdo A., González León M.A., Mateos Guilarte J., Kink manifolds in (1+1)-dimensional scalar field theory, J. Phys. A: Math. Gen. 31 (1998), 209-229.
  11. Alonso Izquierdo A., Mateos Guilarte J., Generalized MSTB models: structure and kink varietes, Phys. D 237 (2008), 3263-3291, arXiv:0802.0153.
  12. Alonso Izquierdo A., González León M.A., Mateos Guilarte J., Kinks from dynamical systems: domain walls in a deformed linear O(N) sigma model, Nonlinearity 13 (2000), 1137-1169, hep-th/0003224.
  13. Neumann C., De problemate quodam mechanico, quod ad primam integralium ultraelipticorum classem revocatur, J. Reine Angew. Math. 56 (1859), 46-63.
  14. Moser J., Various aspects of integrable Hamiltonian systems, in Dynamical Systems (C.I.M.E. Summer School, Bressanone, 1978), Progr. Math., Vol. 8, Birkhäuser, Boston, 1980, 233-289.
  15. Dubrovin B.A., Theta functions and nonlinear equations, Uspekhi Mat. Nauk 36 (1981), no. 2, 11-80.
  16. Alonso Izquierdo A., González León M.A., Mateos Guilarte J., Stability of kink defects in a deformed O(3) linear sigma model, Nonlinearity 15 (2002), 1097-1125, math-ph/0204041.
  17. Bogomolnyi E.B., The stability of classical solutions, Soviet J. Nuclear Phys. 24 (1976), 449-454.
  18. Mumford D., Tata lectures on theta. II. Jacobian theta functions and differential equations, Progr. Math., Vol. 43, Birkhäuser, Boston, 1984.
  19. Ito H., Tasaki H., Stability theory for nonlinear Klein-Gordon kinks and the Morse's index theorem, Phys. Lett. A 113 (1985), 179-182.

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