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

SIGMA 6 (2010), 055, 27 pages      arXiv:1004.1627

Bidifferential Calculus Approach to AKNS Hierarchies and Their Solutions

Aristophanes Dimakis a and Folkert Müller-Hoissen b
a) Department of Financial and Management Engineering, University of the Aegean, 41, Kountourioti Str., GR-82100 Chios, Greece
b) Max-Planck-Institute for Dynamics and Self-Organization, Bunsenstrasse 10, D-37073 Göttingen, Germany

Received April 12, 2010, in final form June 21, 2010; Published online July 16, 2010

We express AKNS hierarchies, admitting reductions to matrix NLS and matrix mKdV hierarchies, in terms of a bidifferential graded algebra. Application of a universal result in this framework quickly generates an infinite family of exact solutions, including e.g. the matrix solitons in the focusing NLS case. Exploiting a general Miura transformation, we recover the generalized Heisenberg magnet hierarchy and establish a corresponding solution formula for it. Simply by exchanging the roles of the two derivations of the bidifferential graded algebra, we recover ''negative flows'', leading to an extension of the respective hierarchy. In this way we also meet a matrix and vector version of the short pulse equation and also the sine-Gordon equation. For these equations corresponding solution formulas are also derived. In all these cases the solutions are parametrized in terms of matrix data that have to satisfy a certain Sylvester equation.

Key words: AKNS hierarchy; negative flows; Miura transformation; bidifferential graded algebra; Heisenberg magnet; mKdV; NLS; sine-Gordon; vector short pulse equation; matrix solitons.

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  1. Dimakis A., Müller-Hoissen F., Bidifferential graded algebras and integrable systems, Discrete Contin. Dyn. Syst. Suppl. 2009 (2009), 208-219, arXiv:0805.4553.
  2. Dimakis A., Müller-Hoissen F., Solutions of matrix NLS systems and their discretisations: a unified treatment, Inverse Problems 26 (2010), 095007, 55 pages, arXiv:1001.0133.
  3. Nijhoff F.W., Linear integral transformations and hierarchies of integrable nonlinear evolution equations, Phys. D 31 (1988), 339-388.
  4. Fuchssteiner B., Fokas A.S., Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D 4 (1981), 47-66.
  5. Verovsky J.M., Negative powers of Olver recursion operators, J. Math. Phys. 32 (1991), 1733-1736.
  6. Tracy C.A., Widom H., Fredholm determinants and the mKdV/sinh-Gordon hierarchies, Comm. Math. Phys. 179 (1996), 1-9, solv-int/9506006.
  7. Ji J., Zhang J.-B., Zhang D.-J., Soliton solutions for a negative order AKNS equation hierarchy, Commun. Theor. Phys. 52 (2009), 395-397.
  8. Dorfmeister J., Gradl H., Szmigielski J., Systems of PDEs obtained from factorization in loop groups, Acta Appl. Math. 53 (1998), 1-58, solv-int/9801009.
  9. Kamchatnov A.M., Pavlov M.V., On generating functions in the AKNS hierarchy, Phys. Lett. A 301 (2002), 269-274, nlin.SI/0208025.
  10. Aratyn H., Ferreira L.A., Gomes J.F., Zimerman A.H., The complex sine-Gordon equation as a symmetry flow of the AKNS hierarchy, J. Phys. A: Math. Gen. 33 (2000), L331-L337, nlin.SI/0007002.
  11. Aratyn H., Gomes J.F., Zimerman A.H., On negative flows of the AKNS hierarchy and a class of deformations of a bihamiltonian structure of hydrodynamic type, J. Phys. A: Math. Gen. 39 (2006), 1099-1114, nlin.SI/0507062.
  12. Hasimoto H., A soliton on a vortex filament, J. Fluid Mech. 51 (1972), 477-485.
  13. Zakharov V.E., Takhtadzhyan L.A., Equivalence of the nonlinear Schrödinger equation and the equation of a Heisenberg ferromagnet, Theoret. and Math. Phys. 38 (1979), 17-23.
  14. Ishimori Y., A relationship between the Ablowitz-Kaup-Newell-Segur and Wadati-Konno-Ichikawa schemes of the inverse scattering method, J. Phys. Soc. Japan 51 (1982), 3036-3041.
  15. Wadati M., Sogo K., Gauge transformations in soliton theory, J. Phys. Soc. Japan 52 (1983), 394-398.
  16. Tsuchida T., Wadati M., Multi-field integrable systems related to WKI-type eigenvalue problems, J. Phys. Soc. Japan 68 (1999), 2241-2245, solv-int/9907018.
  17. Faddeev L.D., Takhtajan L.A., Hamiltonian methods in the theory of solitons, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1987.
  18. van der Linden J., Capel H.W., Nijhoff F.W., Linear integral equations and multicomponent nonlinear integrable systems. II, Phys. A 160 (1989), 235-273.
  19. Gerdjikov V., Grahovski G., On N-wave and NLS type systems: generating operators and the gauge group action: the so(5) case, in Proceedings of Fifth International Conference ''Symmetry in Nonlinear Mathematical Physics'' (June 23-29, 2003, Kyiv), Editors A.G. Nikitin, V.M. Boyko, R.O. Popovych and I.A. Yehorchenko, Proc. Inst. Math. NAS Ukraine, Vol. 50, 2004, Part 1, 388-395.
  20. Zakharov V.E., Shabat A.B., A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I, Funct. Anal. Appl. 8 (1974), 226-235.
  21. Zakharov V., The inverse scattering method, in Solitons, Editors R. Bullough and P. Caudrey, Topics in Current Physics, Vol. 17, Springer, Berlin, 1980, 243-285.
  22. Konopelchenko B.G., On the structure of integrable evolution equations, Phys. Lett. A 79 (1980), 39-43.
  23. Gerdjikov V.S., Grahovski G.G., Kostov N.A., Multicomponent NLS-type equations on symmetric spaces and their reductions, Theoret. and Math. Phys. 144 (2005), 1147-1156.
  24. Gerdjikov V.S., Grahovski G.G., Multi-component NLS models on symmetric spaces: spectral properties versus representation theory, SIGMA 6 (2010), 044, 29 pages, arXiv:1006.0301.
  25. Dimakis A., Müller-Hoissen F., Functional representations of integrable hierarchies, J. Phys. A: Math. Gen. 39 (2006), 9169-9186, nlin.SI/0603018.
  26. Bogdanov L.V., Konopelchenko B.G., Analytic-bilinear approach to integrable hierarchies. II. Multicomponent KP and 2D Toda lattice hierarchies, J. Math. Phys. 39 (1998), 4701-4728, solv-int/9705009.
  27. Konopelchenko B., Strampp W., The AKNS hierarchy as symmetry constraint of the KP hierarchy, Inverse Problems 7 (1991), L17-L24.
  28. Athorne C., Fordy A., Generalised KdV and MKdV equations associated with symmetric spaces, J. Phys. A: Math. Gen. 20 (1987), 1377-1386.
  29. Horn R.A., Johnson C.R., Topics in matrix analysis, Cambridge University Press, Cambridge, 1991.
  30. Cherednik I., Basic methods of soliton theory, Advanced Series in Mathematical Physics, Vol. 25, World Scientific Publishing Co., Inc., River Edge, NJ, 1996.
  31. Golubchik I.Z., Sokolov V.V., Generalized Heisenberg equations on Z-graded Lie algebras, Theoret. and Math. Phys. 120 (1999), 1019-1025.
  32. Rabelo M., On equations which describe pseudospherical surfaces, Stud. Appl. Math. 81 (1989), 221-248.
  33. Beals R., Rabelo M., Tenenblat K., Bäcklund transformations and inverse scattering solutions for some pseudospherical surface equations, Stud. Appl. Math. 81 (1989), 125-151.
  34. Sakovich A., Sakovich S., On transformations of the Rabelo equations, SIGMA 3 (2007), 086, 8 pages, arXiv:0705.2889.
  35. Schäfer T., Wayne C.E., Propagation of ultra-short optical pulses in cubic nonlinear media, Phys. D 196 (2004), 90-105.
  36. Sakovich A., Sakovich S., The short pulse equation is integrable, J. Phys. Soc. Japan 74 (2005), 239-241, nlin.SI/0409034.
  37. Sakovich A., Sakovich S., Solitary wave solutions of the short pulse equation, J. Phys. A: Math. Gen. 39 (2006), L361-L367, nlin.SI/0601019.
  38. Brunelli J.C., The bi-Hamiltonian structure of the short pulse equation, Phys. Lett. A 353 (2006), 475-478, nlin.SI/0601014.
  39. Kuetche V.K., Bouetou T.B., Kofane T.C., On two-loop soliton solution of the Schäfer-Wayne short-pulse equation using Hirota's method and Hodnett-Moloney approach, J. Phys. Soc. Japan 76 (2007), 024004, 7 pages.
  40. Kuetche V.K., Bouetou T.B., Kofane T.C., On exact N-loop soliton solution to nonlinear coupled dispersionless evolution equations, Phys. Lett. A 372 (2008), 665-669.
  41. Parkes E.J., Some periodic and solitary travelling-wave solutions of the short-pulse equation, Chaos Solitons Fractals 38 (2008), 154-159.
  42. Pietrzyk M., Kanattsikov I., Bandelow U., On the propagation of vector ultra-short pulses, J. Nonlinear Math. Phys. 15 (2008), 162-170.
  43. Matsuno Y., Soliton and periodic solutions of the short pulse model equation, arXiv:0912.2576.
  44. Sakovich S., Integrability of the vector short pulse equation, J. Phys. Soc. Japan 77 (2008), 123001, 4 pages, arXiv:0801.3179.

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