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

SIGMA 7 (2011), 096, 48 pages      arXiv:1108.3990      https://doi.org/10.3842/SIGMA.2011.096
Contribution to the Proceedings of the Conference “Symmetries and Integrability of Difference Equations (SIDE-9)”

### Polynomial Bundles and Generalised Fourier Transforms for Integrable Equations on A.III-type Symmetric Spaces

Vladimir S. Gerdjikov a, Georgi G. Grahovski a, b, Alexander V. Mikhailov c and Tihomir I. Valchev a
a) Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 72 Tsarigradsko chausee, Sofia 1784, Bulgaria
b) School of Mathematical Sciences, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland
c) Applied Math. Department, University of Leeds, Woodhouse Lane, Leeds, LS2 9JT, UK

Received May 26, 2011, in final form October 04, 2011; Published online October 20, 2011

Abstract
A special class of integrable nonlinear differential equations related to A.III-type symmetric spaces and having additional reductions are analyzed via the inverse scattering method (ISM). Using the dressing method we construct two classes of soliton solutions associated with the Lax operator. Next, by using the Wronskian relations, the mapping between the potential and the minimal sets of scattering data is constructed. Furthermore, completeness relations for the 'squared solutions' (generalized exponentials) are derived. Next, expansions of the potential and its variation are obtained. This demonstrates that the interpretation of the inverse scattering method as a generalized Fourier transform holds true. Finally, the Hamiltonian structures of these generalized multi-component Heisenberg ferromagnetic (MHF) type integrable models on A.III-type symmetric spaces are briefly analyzed.

Key words: reduction group; Riemann-Hilbert problem; spectral decompositions; integrals of motion.

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References

1. Ablowitz M.J., Kaup D.J., Newell A.C., Segur H., The inverse scattering transform - Fourier analysis for nonlinear problems, Studies in Appl. Math. 53 (1974), 249-315.
2. Adler V.E., Shabat A.B., Yamilov R.I., The symmetry approach to the problem of integrability, Theoret. and Math. Phys. 125 (2000), 1603-1661.
3. Athorne C., Fordy A., Generalised KdV and MKdV equations associated with symmetric spaces, J. Phys. A: Math. Gen. 20 (1987), 1377-1386.
Athorne C., Fordy A., Integrable equations in (2+1) dimensions associated with symmetric and homogeneous spaces, J. Math. Phys. 28 (1987), 2018-2024.
4. Borissov A.B., Kisselev V.V., Nonlinear waves, solitons and localised structures in magnets, Vol. 1, Quasi-homogeneous magnetic solitons, Inst. Metal Physics, Ural Branch of RAS, Ekaterininburg, 2009 (in Russian).
5. Calogero F., Degasperis A., Nonlinear evolution equations solvable by the inverse spectral transform. I, Nuovo Cimento B 32 (1976), 201-242.
6. Calogero F., Degasperis A., Nonlinear evolution equations solvable by the inverse spectral transform. II, Nuovo Cimento B 39 (1976), 1-54.
7. Drinfel'd V.G., Sokolov V.V., Lie algebras and equations of Korteweg-de Vries type, J. Math. Sci. 30 (1985), 1975-2036.
8. Fordy A.P., Derivative nonlinear Schrödinger equations and Hermitian symmetric spaces,  J. Phys. A: Math. Gen. 17 (1984), 1235-1245.
9. Fordy A.P., Kulish P.P., Nonlinear Schrödinger equations and simple Lie algebras, Comm. Math. Phys. 89 (1983), 427-443.
10. Gel'fand I.M., Dickey L.A., Asymptotic behaviour of the resolvent of Sturm-Liouville equations and the algebra of the Korteweg-de Vries equations, Russ. Math. Surv. 30 (1975), no. 5, 77-1113.
Gel'fand I.M., Dickey L.A., A Lie algebra structure in a formal variational calculation, Funct. Anal. Appl. 10 (1976), no. 1, 16-22.
11. Gerdjikov V.S., Generalised Fourier transforms for the soliton equations. Gauge-covariant formulation, Inverse Problems 2 (1986), 51-74.
12. Gerdjikov V.S., The generalized Zakharov-Shabat system and the soliton perturbations, Theoret. and Math. Phys. 99 (1994), 593-598.
13. Gerdjikov V.S., Grahovski G.G., Kostov N.A., Reductions of N-wave interactions related to simple Lie algebras. I. Z2-reductions, J. Phys. A: Math. and Gen. 34 (2001), 9425-9461, nlin.SI/0006001.
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.
14. Gerdjikov V.S., Grahovski G.G., Mikhailov A.V., Valchev T.I., Rational bundles and recursion operators for integrable equations on A.III-type symmetric spaces, Theoret. and Math. Phys., to appear, arXiv:1102.1942.
15. Gerdjikov V.S., Khristov E.Kh., On the evolution equations solvable with the inverse scattering problem. II. Hamiltonian structures and Bäcklund transformations, Bulgar. J. Phys. 7 (1980), 119-133 (in Russian).
16. Gerdjikov V.S., Ivanov M.I., The quadratic bundle of general form and the nonlinear evolution equations. II. Hierarchies of Hamiltonian structures, Bulgar. J. Phys. 10 (1983), 130-143 (in Russian).
17. Gerdjikov V.S., Kulish P.P., The generating operator for the n×n linear system, Phys. D 3 (1981), 549-564.
18. Gerdjikov V.S., Mikhailov A.V., Valchev T.I., Reductions of integrable equations on A.III-type symmetric spaces, J. Phys. A: Math Theor. 43 (2010), 434015, 13 pages, arXiv:1004.4182.
19. Gerdjikov V.S., Mikhailov A.V., Valchev T.I., Recursion operators and reductions of integrable equations on symmetric spaces, J. Geom. Symmetry Phys. 20 (2010), 1-34.
20. Gerdjikov V.S., Vilasi G., Yanovski A.B., Integrable Hamiltonian hierarchies. Spectral and geometric methods, Lecture Notes in Physics, Vol. 748, Springer-Verlag, Berlin, 2008.
21. Golubchik I.Z., Sokolov V.V., Integrable equations on Z-graded Lie algebras, Theoret. and Math. Phys. 112 (1997), 1097-1103.
22. Golubchik I.Z., Sokolov V.V., Generalised Heisenberg equations on Z-graded Lie algebras, Theoret. and Math. Phys. 120 (1999), 1019-1025.
23. Golubchik I.Z., Sokolov V.V., Multicomponent generalization of the hierarchy of the Landau-Lifshitz equation, Theoret. and Math. Phys. 124 (2000), 909-917.
24. Gürses M., Karasu A., Sokolov V.V., On construction of recursion operators from Lax representation, J. Math. Phys. 40 (1999), 6473-6490, solv-int/9909003.
25. Helgasson S., Differential geometry, Lie groups and symmetric spaces, Pure and Applied Mathematics, Vol. 80, Academic Press, Inc., New York - London, 1978.
26. Ibragimov N.Kh., Shabat A.B., Infinite Lie-Bäcklund algebras, Funct. Anal. Appl. 14 (1980), 313-315.
27. Kaup D.J., Newell A.C., An exact solution for a derivative nonlinear Schrödinger equation, J. Math. Phys. 19 (1978), 798-801.
28. Lombardo S., Mikhailov A.V., Reductions of integrable equations: dihedral group, J. Phys. A: Math. Gen. 37 (2004), 7727-7742, nlin.SI/0404013.
29. Lombardo S., Mikhailov A.V., Reduction groups and automorphic Lie algebras, Comm. Math. Phys. 258 (2005), 179-202, math-ph/0407048.
30. Lombardo S., Sanders J.A., On the classification of automorphic Lie algebras, Comm. Math. Phys. 299 (2010), 793-824, arXiv:0912.1697.
31. Loos O., Symmetric spaces, Vol. I, General theory; Vol. II, Compact spaces and classification, W.A. Benjamin, Inc., New York - Amsterdam, 1969.
32. Mikhailov A.V., On the integrability of two-dimensional generalization of the Toda lattice, Lett. J. Exper. and Theor. Phys. 30 (1979), 443-448.
33. Mikhailov A.V., Reduction in integrable systems. The reduction group, Lett. J. Exper. and Theor. Phys. 32 (1980), 187-192.
34. Mikhailov A.V., The reduction problem and the inverse scattering method, Phys. D 3 (1981), 73-117.
35. Mikhailov A.V., The Landau-Lifschitz equation and the Riemann boundary problem on a torus, Phys. Lett. A 92 (1982), 51-55.
36. Mikhailov A.V., Olshanetski M.A., Perelomov A.M., Two-dimensional generalized Toda lattice, Comm. Math. Phys. 79 (1981), 473-488.
37. Mikhailov A.V., Sokolov V.V., Symmetries of differential equations and the problem of integrability, in Integrability, Editor A.V. Mikhailov, Lecture Notes in Physics, Vol. 767, Springer-Verlag, Berlin, 2009, 19-88.
38. Mikhailov A.V., Sokolov V.V., Shabat A.B., The symmetry approach to classification of integrable equations, in What is Integrability?, Editor V.E. Zakharov, Springer Ser. Nonlinear Dynam., Springer-Verlag, Berlin, 1991, 115-184.
39. Mikhailov A.V., Shabat A.B., Yamilov R.I., The symmetry approach to the classification of nonlinear equations. Complete lists of integrable systems, Russ. Math. Surv. 42 (1987), no. 4, 1-63.
40. Novikov S., Manakov S.V., Pitaevskii L.P., Zakharov V.E., Theory of solitons. The inverse scattering method, Plenum, New York, 1984.
41. Olver P.J., Evolution equations possessing infinitely many symmetries, J. Math. Phys. 18 (1977), 1212-1215.
42. Sokolov V.V., Shabat A.B., Classification of integrable evolution equations, Soviet Sci. Rev. Sect. C Math. Phys. Rev., Vol. 4, Harwood Academic Publ., Chur, 1984, 221-280.
43. Svinolupov S.I., Sokolov V.V., Evolution equations with nontrivial conservation laws, Funct. Anal. Appl. 16 (1982), 317-319.
44. Svinolupov S.I., Sokolov V.V., On conservation laws for equations with nontrivial Lie-Bäcklund algebra, in Integrable Systems, Editor A.B. Shabat, Ufa, BFAN SSSR, 1982, 53-67 (in Russian).
45. Takhtadjan L., Faddeev L., The Hamiltonian approach to soliton theory, Springer-Verlag, Berlin, 1986.
46. Wang J.P., Lenard scheme for two-dimensional periodic Volterra chain, J. Math. Phys. 50 (2009), 023506, 25 pages, arXiv:0809.3899.
47. Wu D., Ma H., Twisted hierarchies associated with the generalized sine-Gordon equation, arXiv:1103.6077.
48. Zakharov V.E., Manakov S.V., Exact theory of resonant interaction of wave packets in nonlinear media, INF preprint 74-41, Novosibirsk, 1974, 52 pages (in Russian).
Zakharov V.E., Manakov S.V., On the theory of resonant interaction of wave packets in nonlinear media, Soviet Physics JETP 42 (1975), 842-850.
Zakharov V.E., Manakov S.V., Asymptotic behavior of nonlinear wave systems integrable by the inverse scattering method, Soviet Physics JETP 44 (1976), 106-112 (in Russian).
49. 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. Anal. Appl. 8 (1974), 226-235.
Zakharov V.E., Shabat A.B., Integration of nonlinear equations of mathematical physics by the method of the inverse scattering transform. II, Funct. Anal. Appl. 13 (1979), 166-174.
50. Zakharov V.E., Mikhailov A.V., On the integrability of classical spinor models in two-dimensional space-time, Comm. Math. Phys. 74 (1980), 21-40.
51. Zakharov V.E., Mikhailov A.V., Relativistically invariant two-dimensional models of field theory which are integrable by means of the inverse scattering problem method, Soviet Physics JETP 47 (1978), 1017-1027.