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KdV ’95 pp 405–433Cite as

Darboux Transformations for Higher-Rank Kadomtsev—Petviashvili and Krichever—Novikov Equations

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Abstract

It is shown that the action of a special ‘rank 2’ or ‘rank 3’ Darboux transformation, called transference, upon a pair of commuting ordinary differential operators of orders 4 and 6 implements the Abelian sum on their elliptic joint spectrum. A consequence of this is that, under the deformation of these commuting operators by the KP flow, every ‘rank 2’ KP solution corresponds to a solution of the Krichever—Novikov (KN) equation, and vice versa, with the transference process providing the correspondence between (2 + 1) and (1 + 1) dimensions. For a singular joint spectrum, it is further shown that transference at the singular point produces a correspondence between solutions of the singular KN equation and those of the KdV equation. These correspondences are illustrated by considering examples of a nondecaying rational KdV or Boussinesq solutions and the corresponding rational, singular-KN and rational KP solutions which the transference process generates.

Mathematics Subject Classifications (1991): 58F07.

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References

  1. Adler, M. and Moser,J.: On a class of polynomials connectedwith the Korteweg-de Vriesequations, Comm. Math. Phys. 61 (1978), 1–30.

    Article  MathSciNet  MATH  Google Scholar 

  2. Burchnall, J.L. and Chaundy, T. W.: Commutative ordinary differential operators, Proc. London Math. Soc. 211 (1923),420–440.

    Article  MathSciNet  Google Scholar 

  3. Burchnall, J.L. and Chaundy, T. W.: Commutative ordinary differential operators, Proc.Royal Soc. London (A) 118 (1928),557–583.

    Article  MATH  Google Scholar 

  4. Burchnall, J. L. and Chaundy, T.W.: Commutative ordinary differential operators, II. Theidentity P n = Q m, Proc. Royal Soc. London (A) 134(1932), 471–485.

    Article  Google Scholar 

  5. Cherednik, I. V.: Differentialequations for the Baker-Akhiezer functions of algebraic curves,Funct. Anal. Appl. 12 (1978), 195–203.

    Article  MathSciNet  Google Scholar 

  6. Dehornoy, P.: Opérateurs différentielset courbes elliptiques, Compositio Math. 43 (1981),71–99.

    MathSciNet  MATH  Google Scholar 

  7. Dixmier, J.: Sur les Algebras de Weyl, Bull.Soc. Math. France 96 (1968), 209–242.

    MathSciNet  MATH  Google Scholar 

  8. Dorfman, I.Ya. and Nijhoff, F. W.: On a (2+1)-dimensional version of the Krichever-Novikovequation, Clarkson University, PreprintINS, No. 167.

    Google Scholar 

  9. Drinfeld, V. G. and Sokolov, V. V.:On equations related to the Korteweg-de Vries equation,Soviet Math. Dokl. 32 (1985), 361–365.

    Google Scholar 

  10. Duistermaat,J. J. and Grünbaum, F. A.: Differential equations in thespectral parameter, Comm.Math. Phys. 103 (1986),177–240.

    Article  MathSciNet  MATH  Google Scholar 

  11. Ehlers, F.and Knörrer, H.: Analgebro-geometric interpretation of the Bäcklund transformationfor the Korteweg-de Vries equation, Comment.Math. Helvetici 57 (1982), 1–10.

    Article  MATH  Google Scholar 

  12. Grinevich, G.P. G.: Rational solutions for the equation of commutation of differentialoperators,Funct. Anal. Appl. 16 (1982), 19–24.

    Article  MathSciNet  MATH  Google Scholar 

  13. Grünbaum, F. A.: Commuting pairs of linearordinary differential operators of orders four andsix, Physica 31D (1988),424–433.

    Google Scholar 

  14. Grünbaum, F. A.: TheKadomtsev-Petviashvili equation: an elementary approach to the ‘rank2’ solutions of Krichever and Novikov, Phys. Lett. A 139 (1989),146–150.

    Article  MathSciNet  Google Scholar 

  15. Grünbaum, F. A.: Darboux’s method and some ‘rank two’ explicit solutions of the KP equation,in A. Degasperis, A. P. Fordy and M. Lakshmanan (eds), Nonlinear Evolution Equations:Integrability and Spectral Methods,Manchester University Press,Manchester, 1990, pp. 271–277.

    Google Scholar 

  16. Ince, E. L.: Ordinary Differential Equations, Dover Publications, New York, 1956.

    Google Scholar 

  17. Krichever, I. M.: Commutative ringsof ordinary differential operators, Funct. Anal. Appl. 12(1978), 175–185.

    Article  Google Scholar 

  18. Krichever, I. M. and Novikov, S.P.: Holomorphic fiberings and nonlinear equations, finitezone solutions of rank 2, Sov. Math. Dokl. 20 (1979), 650–654.

    MATH  Google Scholar 

  19. Krichever, I.M. and Novikov, S. P.: Holomorphic bundles over algebraic curves and nonlinearequations, Soviet Math. Surveys 35(1980), 53–79.

    Article  MathSciNet  MATH  Google Scholar 

  20. Latham, G. A.: Solutions of the KP equationassociated to rank 3 commuting differentialoperators over a singular elliptic curve, Physica D41 (1990),55–66.

    MathSciNet  Google Scholar 

  21. Latham, G. A.: The computationalapproach to commuting ordinary differential operators oforders six and nine, J. Austral. Math. Soc, Series B 35 (1994),399–419.

    Article  MathSciNet  MATH  Google Scholar 

  22. Latham, G. A. and Previato, E.:Higher rank Darboux transformations, in N. M. Ercolani, I.R. Gabitov, C. D. Levermore and D. Serre(eds), Singular Limits of Dispersive Waves, NATOASI Series, Series B: Physics, Vol. 320, Plenum, New York, 1994, pp.117–134.

    Chapter  Google Scholar 

  23. Latham, G. A. and Previato, E.: KP solutionsgenerated from KdV by ‘rank 2’ transference,submitted to Physica D.

    Google Scholar 

  24. Mokhov, I. O.: Commutingdifferential operators of rank 3 and nonlinear equations, Math.USSR Izvestiya 35 (1990), 629–655.

    Article  MathSciNet  MATH  Google Scholar 

  25. Mumford, D.: An algebro-geometricconstruction of commuting operators and of solutions tothe Toda lattice equation, KdV equation and related nonlinear equations, in M.Nagata (ed.),Proc. Int. Symp. Alg. Geom. Kinokuniya, Tokyo, 1978, pp. 115–153.

    Google Scholar 

  26. Previato, E.: 70 years of spectralcurves: 1923–1993, in Proc. CIME 1993, Lecture Notes inPhysics, Springer-Verlag (to appear).

    Google Scholar 

  27. Previato, E. and Wilson, G.: Vectorbundles over curves and solutions of the KP equations, in Proc. Sympos. Pure Math. 49 (1989), 553–569.

    MathSciNet  Google Scholar 

  28. Previato, E. and Wilson, G.:Differential operators and rank two bundles over elliptic curves,Compositio Math. 81 (1992), 107–119.

    MathSciNet  MATH  Google Scholar 

  29. Segal, G. B. and Wilson, G.: Loopgroups and equations of KdV type, Publ Math. I.H.E.S.61 (1985), 5–65.

    MathSciNet  MATH  Google Scholar 

  30. Sokolov, V.V.: On the Hamiltonian property of the Krichever-Novikov equation, Dokl.Akad.Nauk SSSR 277 (1984), 48–50

    MathSciNet  Google Scholar 

  31. Sokolov, V.V. On the Hamiltonian property of the Krichever-Novikov equation. Sov. Math.Dokl. 30 (1984), 44–46.

    MATH  Google Scholar 

  32. Svinolupov,S. I., Sokolov, V, and Yamilov, R.: On Bäcklund transformations for integrableevolution equations, Soviet Math. Dokl.28 (1983), 165–168.

    MATH  Google Scholar 

  33. Wilson, W. G.: Algebraic curves andsoliton equations, in E. Arbarello, C. Procesi and E.Strickland (eds), Geometry Today, Birkhäuser, Boston, 1985, pp. 303–329.

    Google Scholar 

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Authors and Affiliations

  1. Burdett, Buckeridge and Young Ltd., Level 45, Rialto South Tower, 525 Collins St., Melbourne, Vic, 3000, Australia

    Geoff A. Latham

  2. Department of Mathematics, Boston University, Boston, MA, 02215, USA

    Emma Previato

Authors
  1. Geoff A. Latham
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  2. Emma Previato
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Michiel Hazewinkel Hans W. Capel Eduard M. de Jager

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© 1995 Springer Science+Business Media Dordrecht

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Latham, G.A., Previato, E. (1995). Darboux Transformations for Higher-Rank Kadomtsev—Petviashvili and Krichever—Novikov Equations. In: Hazewinkel, M., Capel, H.W., de Jager, E.M. (eds) KdV ’95. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-0017-5_23

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  • DOI: https://doi.org/10.1007/978-94-011-0017-5_23

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-4011-2

  • Online ISBN: 978-94-011-0017-5

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