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


SIGMA 3 (2007), 039, 19 pages      nlin.SI/0703002      https://doi.org/10.3842/SIGMA.2007.039
Contribution to the Vadim Kuznetsov Memorial Issue

N-Wave Equations with Orthogonal Algebras: Z2 and Z2 × Z2 Reductions and Soliton Solutions

Vladimir S. Gerdjikov a, Nikolay A. Kostov a, b and Tihomir I. Valchev a
a) Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 72 Tsarigradsko chaussee, 1784 Sofia, Bulgaria
b) Institute of Electronics, Bulgarian Academy of Sciences, 72 Tsarigradsko chaussee, 1784 Sofia, Bulgaria

Received November 21, 2006, in final form February 08, 2007; Published online March 03, 2007

Abstract
We consider N-wave type equations related to the orthogonal algebras obtained from the generic ones via additional reductions. The first Z2-reduction is the canonical one. We impose a second Z2-reduction and consider also the combined action of both reductions. For all three types of N-wave equations we construct the soliton solutions by appropriately modifying the Zakharov-Shabat dressing method. We also briefly discuss the different types of one-soliton solutions. Especially rich are the types of one-soliton solutions in the case when both reductions are applied. This is due to the fact that we have two different configurations of eigenvalues for the Lax operator L: doublets, which consist of pairs of purely imaginary eigenvalues, and quadruplets. Such situation is analogous to the one encountered in the sine-Gordon case, which allows two types of solitons: kinks and breathers. A new physical system, describing Stokes-anti Stokes Raman scattering is obtained. It is represented by a 4-wave equation related to the B2 algebra with a canonical Z2 reduction.

Key words: solitons; Hamiltonian systems.

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References

  1. Ackerhalt J.R., Milonni P.W., Solitons and four-wave mixing, Phys. Rev. A 33 (1986), 3185-3198.
  2. Armstrong J., Bloembergen N., Ducuing J., Persham P., Interactions between light waves in a nonlinear dielectric, Phys. Rev. 127 (1962), 1918-1939.
  3. Degasperis A., Lombardo S., Multicomponent integrable wave equations: I. Darboux-dressing transformation, J. Phys. A: Math. Theor. 40 (2007), 961-977, nlin.SI/0610061.
  4. Ferapontov E., Isoparametric hypersurfaces in spheres, integrable nondiagonalizable systems of hydrodynamic type and N-wave systems, Differential Geom. Appl. 5 (1995), 335-369.
  5. Gerdjikov V., On the spectral theory of the integro-differential operator L, generating nonlinear evolution equations, Lett. Math. Phys. 6 (1982), 315-324.
  6. Gerdjikov V., Generalised Fourier transforms for the soliton equations. Gauge covariant formulation, Inverse Problems 2 (1986), 51-74.
  7. Gerdjikov V., Algebraic and analytic aspects of soliton type equations, Contemp. Math. 301 (2002), 35-67, nlin.SI/0206014.
  8. Gerdjikov V., Grahovski G., Ivanov R., Kostov N., 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., Grahovski G., Kostov N., Reductions of N-wave interactions related to low-rank simple Lie algebras, J. Phys A: Math. Gen. 34 (2001), 9425-9461, nlin.SI/0006001.
  10. Gerdjikov V., Kaup D., How many types of soliton solutions do we know? In Proceedings of Seventh International Conference on Geometry, Integrability and Quantization (June 2-10, 2005, Varna, Bulgaria), Editors I. Mladenov and M. de Leon, Softex, Sofia, 2006, 11-34.
  11. Gerdjikov V., Kostov N., Inverse scattering transform analysis of Stokes-anti Stokes stimulated Raman scattering, Phys. Rev. A 54 (1996), 4339-4350, patt-sol/9502001.
  12. Gerdjikov V., Valchev T., Breather solutions of N-wave equations, in Proceedings of Eighth International Conference on Geometry, Integrability and Quantization (June 9-14, 2006, Varna, Bulgaria), Editors I. Mladenov and M. de Leon, to appear.
  13. Goto M., Grosshans F., Semisimple Lie algebras, Marcel Dekker, New York, 1978.
  14. Ivanov R., On the dressing method for the generalised Zakharov-Shabat system, Nuclear Phys. B 694 (2004), 509-524, math-ph/0402031.
  15. Matveev V., Salle M., Darboux transformations and solitons, Springer Verlag, Berlin, 1991.
  16. Mikhailov A., The Reduction problem and the inverse scattering method, Phys. D 3 (1981), 73-117.
  17. Rogers C., Schief W.K., Bäcklund and Darboux transformations, Cambridge University Press, London, 2002.
  18. Rogers C., Shadwick W., Bäcklund transformations and their applications, Academic Press, New York, 1982.
  19. Shabat A., The inverse scattering problem for a system of differential equations, Funktsional. Anal. i Prilozhen. 9, (1975), no. 3, 75-78 (in Russian).
  20. Shabat A., The inverse scattering problem, Differ. Uravn. 15 (1979), 1824-1834 (in Russian).
  21. Takhtadjan L., Faddeev L., The Hamiltonian approach to soliton theory, Springer Verlag, Berlin, 1987.
  22. Zakharov V., Mikhailov A., On the integrability of classical spinor models in two-dimensional space-time, Comm. Math. Phys. 74 (1980), 21-40.
  23. Zakharov V., Manakov S., On the theory of resonant interactions of wave packets in nonlinear media, JETP 69 (1975), no. 5, 1654-1673.
  24. Zakharov V., Manakov S., Novikov S., Pitaevskii L., Theory of solitons: the inverse scattering method, Plenum, New York, 1984.
  25. Zakharov V., Shabat A., Integration of the nonlinear equations of mathematical physics by the inverse scattering method, Funktsional. Anal. i Prilozhen. 13 (1979), no. 3, 13-22 (in Russian).

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