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

SIGMA 4 (2008), 029, 30 pages      arXiv:0803.1651      https://doi.org/10.3842/SIGMA.2008.029
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics

Reductions of Multicomponent mKdV Equations on Symmetric Spaces of DIII-Type

Vladimir S. Gerdjikov and Nikolay A. Kostov
Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 72 Tsarigradsko chaussee, 1784 Sofia, Bulgaria

Received December 14, 2007, in final form February 27, 2008; Published online March 11, 2008

Abstract
New reductions for the multicomponent modified Korteweg-de Vries (MMKdV) equations on the symmetric spaces of DIII-type are derived using the approach based on the reduction group introduced by A.V. Mikhailov. The relevant inverse scattering problem is studied and reduced to a Riemann-Hilbert problem. The minimal sets of scattering data Ti, i = 1, 2 which allow one to reconstruct uniquely both the scattering matrix and the potential of the Lax operator are defined. The effect of the new reductions on the hierarchy of Hamiltonian structures of MMKdV and on Ti are studied. We illustrate our results by the MMKdV equations related to the algebra g @ so(8) and derive several new MMKdV-type equations using group of reductions isomorphic to Z2, Z3, Z4.

Key words: multicomponent modified Korteweg-de Vries (MMKdV) equations; reduction group; Riemann-Hilbert problem; Hamiltonian structures.

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References

  1. Wadati M., The modified Korteweg-de Vries equation, J. Phys. Soc. Japan 34 (1972), 1289-1296.
  2. Athorne C., Fordy A., Generalised KdV and MKDV equations associated with symmetric spaces, J. Phys. A: Math. Gen. 20 (1987), 1377-1386.
  3. Fordy A.P., Kulish P.P., Nonlinear Schrödinger equations and simple Lie algebras, Comm. Math. Phys. 89 (1983), 427-443.
  4. Ablowitz M.J., Prinari B., Trubatch A.D., Discrete and continuous nonlinear Schrödinger systems, London Mathematical Society Lecture Notes Series, Vol. 302, Cambridge University Press, 2004.
  5. Mikhailov A.V., The reduction problem and the inverse scattering problem, Phys. D 3 (1981), 73-117.
  6. Lombardo S., Mikhailov A.V., Reductions of integrable equations. Dihedral group, J. Phys. A: Math. Gen. 37 (2004), 7727-7742, nlin.SI/0404013.
    Lombardo S., Mikhailov A.V., Reduction groups and automorphic Lie algebras, Comm. Math. Phys. 258 (2005), 179-202, math-ph/0407048.
  7. Gerdjikov V.S., Grahovski G.G., Kostov N.A., Reductions of N-wave interactions related to low-rank simple Lie algebras. I. Z2-reductions, J. Phys. A: Math. Gen. 34 (2001), 9425-9461, nlin.SI/0006001.
  8. Gerdjikov V.S., Grahovski G.G., Ivanov R.I., Kostov N.A., N-wave interactions related to simple Lie algebras. Z2-reductions and soliton solutions, Inverse Problems 17 (2001), 999-1015, nlin.SI/0009034.
  9. Gerdjikov V.S., Grahovski G.G., Kostov N.A., On the multi-component NLS-type models and their gauge equivalent, in Proceedings of the International Conference ''Contemporary Aspects of Astronomy, Theoretical and Gravitational Physics'' (May 20-22, 2004, Sofia), Heron Press, Sofia, 2005, 306-317.
  10. Gerdjikov V.S., Grahovski G.G., Kostov N.A., On the multi-component NLS-type equations on symmetric spaces: reductions and soliton solutions, in Proceedings of the Sixth International Conference ''Geometry, Integrability and Quantization'' (July 3-10, 2004, Varna, Bulgaria), Softex, Sofia, 2005, 203-217.
  11. Drinfel'd V., Sokolov V.V., Lie algebras and equations of Korteweg-de Vries type, Sov. J. Math. 30 (1985), 1975-2036.
  12. Helgasson S., Differential geometry, Lie groups and symmetric spaces, Academic Press, 1978.
  13. Zakharov V.E., Shabat A.B., A scheme for integrating nonlinear equations of mathematical physics by the method of the inverse scattering transform. I, Funct. Annal. Appl. 8 (1974), 226-235.
    Zakharov V.E., Shabat A.B., A scheme for integrating nonlinear equations of mathematical physics by the method of the inverse scattering transform. II, Funct. Anal. Appl. 13 (1979), no. 3, 13-23.
  14. Shabat A.B., Inverse-scattering problem for a system of differential equations, Funct. Anal. Appl. 9 (1975), 244-247.
  15. Shabat A.B., An inverse scattering problem, Differ. Equations 15 (1979), 1299-1307.
  16. Zakharov V.E., Manakov S.V., Novikov S.P., Pitaevskii L.I., Theory of solitons: the inverse scattering method, Plenum, New York, 1984.
  17. Calogero F., Degasperis A., Nonlinear evolution equations solvable by the inverse spectral transform. I, Nuovo Cimento B (11) 32 (1976), 201-242.
    Calogero F., Degasperis A., Nonlinear evolution equations solvable by the inverse spectral transform. II, Nuovo Cimento B (11) 39 (1976), 1-54.
  18. Calogero F., Degasperis A., Spectral transform and solitons, Vol. I. North Holland, Amsterdam, 1982.
  19. Faddeev L.D., Takhtadjan L.A., Hamiltonian methods in the theory of solitons, Springer Verlag, Berlin, 1987.
  20. Calogero F., Degasperis A., Coupled nonlinear evolution equations solvable via the inverse spectral transform, and solitons that come back: the boomeron, Lett. Nuovo Cimento (2) 16 (1976), 425-433.
  21. Degasperis A., Solitons, boomerons, trappons, in Nonlinear Evolution Equations Solvable by the Spectral Transform, Editor F. Calogero, Pitman, London, 1978, 97-126.
  22. Beals R., Sattinger D.H., On the complete integrability of completely integrable systems, Comm. Math. Phys. 138 (1991), 409-436.
  23. Gerdjikov V.S., On the spectral theory of the integro-differential operator L, generating nonlinear evolution equations, Lett. Math. Phys. 6 (1982), 315-324.
  24. Gakhov F.D., Boundary value problems, Oxford, Pergamon Press, 1966 (translated from Russian, Editor I.N. Sneddon).
  25. Manakov S.V., On the theory of two-dimensional stationary self-focusing of electromagnetic waves, Zh. Eksp. Teor. Fiz. 65 (1973), 505-516 (English transl.: Sov. Phys. J. Exp. Theor. Phys. 38 (1974), 248-253).
  26. Gerdjikov V.S., Algebraic and analytic aspects of soliton type equations, Contemp. Math. 301 (2002), 35-68, nlin.SI/0206014.
  27. Ablowitz M., Kaup D., Newell A., Segur H., The inverse scattering transform - Fourier analysis for nonlinear problems, Studies in Appl. Math. 53 (1974), 249-315.
  28. Kaup D.J., Newell A.C., Soliton equations, singular dispersion relations and moving eigenvalues, Adv. Math. 31 (1979), 67-100.
  29. Gerdjikov V.S., Generalized Fourier transforms for the soliton equations. Gauge covariant formulation, Inverse Problems 2 (1986), 51-74.
    Gerdjikov V.S., Generating operators for the nonlinear evolution equations of soliton type related to the semisimple Lie algebras, Doctor of Sciences Thesis, JINR, Dubna, USSR, 1987 (in Russian).
  30. Gerdjikov V.S., The Zakharov-Shabat dressing method amd the representation theory of the semisimple Lie algebras, Phys. Lett. A 126 (1987), 184-188.

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