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


SIGMA 12 (2016), 087, 17 pages      arXiv:1512.02386      https://doi.org/10.3842/SIGMA.2016.087
Contribution to the Special Issue on Analytical Mechanics and Differential Geometry in honour of Sergio Benenti

Bäcklund Transformations and Non-Abelian Nonlinear Evolution Equations: a Novel Bäcklund Chart

Sandra Carillo ab, Mauro Lo Schiavo a and Cornelia Schiebold cd
a) Dipartimento ''Scienze di Base e Applicate per l'Ingegneria'', Sapienza - Università di Roma, 16, Via A. Scarpa, 00161 Rome, Italy
b) I.N.F.N. - Sez. Roma1, Gr. IV - Mathematical Methods in NonLinear Physics, Rome, Italy
c) Department of Science Education and Mathematics, Mid Sweden University, S-851 70 Sundsvall, Sweden
d) Instytut Matematyki, Uniwersytet Jana Kochanowskiego w Kielcach, Poland

Received December 08, 2015, in final form August 24, 2016; Published online August 30, 2016

Abstract
Classes of third order non-Abelian evolution equations linked to that of Korteweg-de Vries-type are investigated and their connections represented in a non-commutative Bäcklund chart, generalizing results in [Fuchssteiner B., Carillo S., Phys. A 154 (1989), 467-510]. The recursion operators are shown to be hereditary, thereby allowing the results to be extended to hierarchies. The present study is devoted to operator nonlinear evolution equations: general results are presented. The implied applications referring to finite-dimensional cases will be considered separately.

Key words: .

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References

  1. Aden H., Carl B., On realizations of solutions of the KdV equation by determinants on operator ideals, J. Math. Phys. 37 (1996), 1833-1857.
  2. Athorne C., Fordy A., Generalised KdV and MKdV equations associated with symmetric spaces, J. Phys. A: Math. Gen. 20 (1987), 1377-1386.
  3. Bateman H., The lift and drag functions for an elastic fluid in two-dimensional irrotational flow, Proc. Nat. Acad. Sci. USA 24 (1938), 246-251.
  4. Bateman H., The transformation of partial differential equations, Quart. Appl. Math. 1 (1944), 281-296.
  5. Calogero F., Degasperis A., Nonlinear evolution equations solvable by the inverse spectral transform. II, Nuovo Cimento B 39 (1977), 1-54.
  6. Calogero F., Degasperis A., Spectral transform and solitons. Vol. I. Tools to solve and investigate nonlinear evolution equations, Studies in Mathematics and its Applications, Vol. 13, North-Holland Publishing Co., Amsterdam - New York, 1982.
  7. Carillo S., Nonlinear evolution equations: Bäcklund transformations and Bäcklund charts, Acta Appl. Math. 122 (2012), 93-106.
  8. Carillo S., Fuchssteiner B., The abundant symmetry structure of hierarchies of nonlinear equations obtained by reciprocal links, J. Math. Phys. 30 (1989), 1606-1613.
  9. Carillo S., Lo Schiavo M., Schiebold C., Recursion operators admitted by non-Abelian Burgers equations: some remarks, arXiv:1606.07270.
  10. Carillo S., Rogers C., Bäcklund charts for the Caudrey-Dodd-Gibbon and Kaup-Kupershmidt hierarchies, in Nonlinear Evolutions (Balaruc-les-Bains, 1987), World Sci. Publ., Teaneck, NJ, 1988, 57-73.
  11. Carillo S., Schiebold C., Noncommutative Korteweg-de Vries and modified Korteweg-de Vries hierarchies via recursion methods, J. Math. Phys. 50 (2009), 073510, 14 pages.
  12. Carillo S., Schiebold C., A non-commutative operator-hierarchy of Burgers equations and Bäcklund transformations, in Applied and Industrial Mathematics in Italy III, Ser. Adv. Math. Appl. Sci., Vol. 82, World Sci. Publ., Hackensack, NJ, 2010, 175-185.
  13. Carillo S., Schiebold C., Matrix Korteweg-de Vries and modified Korteweg-de Vries hierarchies: noncommutative soliton solutions, J. Math. Phys. 52 (2011), 053507, 21 pages.
  14. Carillo S., Schiebold C., On the recursion operator for the noncommutative Burgers hierarchy, J. Nonlinear Math. Phys. 19 (2012), 1250003, 11 pages.
  15. Carl B., Schiebold C., Nonlinear equations in soliton physics and operator ideals, Nonlinearity 12 (1999), 333-364.
  16. Carl B., Schiebold C., Ein direkter Ansatz zur Untersuchung von Solitonengleichungen, Jahresber. Deutsch. Math.-Verein. 102 (2000), 102-148.
  17. Cole J.D., On a quasi-linear parabolic equation occurring in aerodynamics, Quart. Appl. Math. 9 (1951), 225-236.
  18. Depireux D.A., Schiff J., On UrKdV and UrKP, Lett. Math. Phys. 33 (1995), 99-111, solv-int/9402004.
  19. Fokas A.S., Fuchssteiner B., Bäcklund transformations for hereditary symmetries, Nonlinear Anal. 5 (1981), 423-432.
  20. Fuchssteiner B., Application of hereditary symmetries to nonlinear evolution equations, Nonlinear Anal. 3 (1979), 849-862.
  21. Fuchssteiner B., The Lie algebra structure of degenerate Hamiltonian and bi-Hamiltonian systems, Progr. Theoret. Phys. 68 (1982), 1082-1104.
  22. Fuchssteiner B., Solitons in interaction, Progr. Theoret. Phys. 78 (1987), 1022-1050.
  23. Fuchssteiner B., Carillo S., Soliton structure versus singularity analysis: third-order completely integrable nonlinear differential equations in $1+1$-dimensions, Phys. A 154 (1989), 467-510.
  24. Fuchssteiner B., Carillo S., The action-angle transformation for soliton equations, Phys. A 166 (1990), 651-675.
  25. Fuchssteiner B., Chowdhury A.R., A new approach to the quantum KdV, Chaos Solitons Fractals 5 (1995), 2345-2355.
  26. Fuchssteiner B., Fokas A.S., Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D 4 (1981), 47-66.
  27. Fuchssteiner B., Oevel W., The bi-Hamiltonian structure of some nonlinear fifth- and seventh-order differential equations and recursion formulas for their symmetries and conserved covaria, J. Math. Phys. 23 (1982), 358-363.
  28. Fuchssteiner B., Schulze T., Carillo S., Explicit solutions for the Harry Dym equation, J. Phys. A: Math. Gen. 25 (1992), 223-230.
  29. Gu C., Hu H., Zhou Z., Darboux transformations in integrable systems. Theory and their applications to geometry, Mathematical Physics Studies, Vol. 26, Springer, Dordrecht, 2005.
  30. Guo B.Y., Carillo S., Infiltration in soils with prescribed boundary concentration, Acta Math. Appl. Sinica 6 (1990), 365-369.
  31. Guo B.Y., Rogers C., On Harry-Dym equation and its solution, Sci. China Ser. A 32 (1989), 283-295.
  32. 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.
  33. Hopf E., The partial differential equation $u_t+uu_x=\mu u_{xx}$, Comm. Pure Appl. Math. 3 (1950), 201-230.
  34. Khalilov F.A., Khruslov E.Ya., Matrix generalisation of the modified Korteweg-de Vries equation, Inverse Problems 6 (1990), 193-204.
  35. Kupershmidt B.A., On a group of automorphisms of the noncommutative Burgers hierarchy, J. Nonlinear Math. Phys. 12 (2005), 539-549.
  36. Levi D., Ragnisco O., Bruschi M., Continuous and discrete matrix Burgers' hierarchies, Nuovo Cimento B 74 (1983), 33-51.
  37. Liu Q.P., Athorne C., Comment on ''Matrix generalisation of the modified Korteweg-de Vries equation'', Inverse Problems 7 (1991), 783-785.
  38. Magri F., A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19 (1978), 1156-1162.
  39. Marchenko V.A., Nonlinear equations and operator algebras, Mathematics and its Applications (Soviet Series), Vol. 17, D. Reidel Publishing Co., Dordrecht, 1988.
  40. Oevel W., Carillo S., Squared eigenfunction symmetries for soliton equations. I, J. Math. Anal. Appl. 217 (1998), 161-178.
  41. Oevel W., Carillo S., Squared eigenfunction symmetries for soliton equations. II, J. Math. Anal. Appl. 217 (1998), 179-199.
  42. Oevel W., Rogers C., Gauge transformations and reciprocal links in $2+1$ dimensions, Rev. Math. Phys. 5 (1993), 299-330.
  43. Olver P.J., Evolution equations possessing infinitely many symmetries, J. Math. Phys. 18 (1977), 1212-1215.
  44. Olver P.J., Sokolov V.V., Integrable evolution equations on associative algebras, Comm. Math. Phys. 193 (1998), 245-268.
  45. Rogers C., Reciprocal relations in non-steady one-dimensional gasdynamics, Z. Angew. Math. Phys. 19 (1968), 58-63.
  46. Rogers C., Ames W.F., Nonlinear boundary value problems in science and engineering, Mathematics in Science and Engineering, Vol. 183, Academic Press, Inc., Boston, MA, 1989.
  47. Rogers C., Carillo S., On reciprocal properties of the Caudrey-Dodd-Gibbon and Kaup-Kupershmidt hierarchies, Phys. Scripta 36 (1987), 865-869.
  48. Rogers C., Schief W.K., Bäcklund and Darboux transformations. Geometry and modern applications in soliton theory, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002.
  49. Rogers C., Shadwick W.F., Bäcklund transformations and their applications, Mathematics in Science and Engineering, Vol. 161, Academic Press, Inc., New York - London, 1982.
  50. Schiebold C., From the non-abelian to the scalar two-dimensional Toda lattice, Glasg. Math. J. 47 (2005), 177-189.
  51. Schiebold C., Noncommutative AKNS systems and multisoliton solutions to the matrix sine-Gordon equation, Discrete Contin. Dyn. Syst. 2009 (2009), 678-690.
  52. Schiebold C., Cauchy-type determinants and integrable systems, Linear Algebra Appl. 433 (2010), 447-475.
  53. Schiebold C., The noncommutative AKNS system: projection to matrix systems, countable superposition and soliton-like solutions, J. Phys. A: Math. Theor. 43 (2010), 434030, 18 pages.
  54. Schiebold C., Structural properties of the noncommutative KdV recursion operator, J. Math. Phys. 52 (2011), 113504, 16 pages.
  55. Schiff J., Symmetries of KdV and loop groups, solv-int/9606004.
  56. Svinolupov S.I., Sokolov V.V., Vector-matrix generalizations of classical integrable equations, Theoret. and Math. Phys. 100 (1994), 959-962.
  57. Weiss J., On classes of integrable systems and the Painlevé property, J. Math. Phys. 25 (1984), 13-24.
  58. Wilson G., On the quasi-Hamiltonian formalism of the KdV equation, Phys. Lett. A 132 (1988), 445-450.

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