Geodesic
Necessary Covariance Conditions for a One-Field Lax Pair
Teoretičeskaâ i matematičeskaâ fizika, Tome 144 (2005) no. 1, pp. 122-132 Cet article a éte moissonné depuis la source Math-Net.Ru

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We study the covariance with respect to Darboux transformations of polynomial differential and difference operators with coefficients given by functions of one basic field. In the scalar (Abelian) case, the functional dependence is established by equating the Frechet differential (the first term of the Taylor series on the prolonged space) to the Darboux transform; a Lax pair for the Boussinesq equation is considered. For a pair of generalized Zakharov–Shabat problems (with differential and shift operators) with operator coefficients, we construct a set of integrable nonlinear equations together with explicit dressing formulas. Non-Abelian special functions are fixed as the fields of the covariant pairs. We introduce a difference Lax pair, a combined gauge-Darboux transformation, and solutions of the Nahm equations.
Mots-clés : Darboux transformation, Lax pair, Boussinesq equation, Zakharov–Shabat problem, Nahm equation.
Keywords: shift operator polynomial 
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S. B. Leble. Necessary Covariance Conditions for a One-Field Lax Pair. Teoretičeskaâ i matematičeskaâ fizika, Tome 144 (2005) no. 1, pp. 122-132. http://geodesic.mathdoc.fr/item/TMF_2005_144_1_a12/

[1] V. B. Matveev, Lett. Math. Phys., 3 (1979), 213 ; 217 ; 503 | DOI | MR | Zbl | Zbl | Zbl

[2] A. A. Zaitsev, S. B. Leble, Rep. Math. Phys., 46 (2000), 165 ; E-print math-ph/9903005 | DOI | MR | Zbl

[3] F. di Bruno, Pure Appl. Math., 1 (1857), 359

[4] S. Leble, “Darboux transforms algebras in $2+1$ dimensions”, Nonlinear Evolution Equations and Dynamical Systems, Proc. of the 7th Workshop (NEEDS-91) (Baia, Verde, Italy, June 19–29, 1991), eds. M. Boiti, L. Martina, F. Pempinelli, World Scientific, Singapore, 1992, 53 | MR | Zbl

[5] S. B. Leble, TMF, 128 (2001), 65 | DOI | MR | Zbl

[6] F. Lambert, S. Leble, J. Springael, Glasgow Math. J. A, 43 (2001), 55 | MR

[7] F. Lambert, I. Loris, J. Springael, Inverse Problems, 17 (2001), 1067 | DOI | MR | Zbl

[8] M. A. Sall, TMF, 53 (1982), 227 ; С. Б. Лебле, М. А. Салль, ДАН СССР, 284 (1985), 110 | MR | Zbl | MR

[9] V. B. Matveev, M. A. Salle, Darboux transformations and solitons, Springer, Berlin, 1991 | MR

[10] C. Roger, W. K. Schiff, Backlund and Darboux Transformations, Camb. Univ. Press, Cambridge, 2002 | MR | Zbl

[11] V. B. Matveev, Am. Math. Soc. Transl. II, 201 (2000), 179 | Zbl

[12] P. J. Olver, V. V. Sokolov, Commun. Math. Phys., 193:2 (1998), 245 | DOI | MR | Zbl

[13] A. V. Mikhailov, V. V. Sokolov, TMF, 122 (2000), 88 | DOI | Zbl

[14] R. H. Heredero, A. Shabat, V. Sokolov, J. Phys. A, 36:47 (2003), L605 | DOI | MR | Zbl

[15] A. B. Shabat, TMF, 136:2 (2003), 197 | DOI | MR | Zbl

[16] S. B. Leble, M. Czachor, Phys. Rev. E, 58 (1998), 7091 | DOI | MR

[17] A. A. Andrianov, F. Cannata, M. V. Ioffe, D. N. Nishnianidze, J. Phys. A, 30 (1997), 5037 | DOI | MR | Zbl

[18] N. Ustinov, S. Leble, M. Czachor, M. Kuna, Phys. Lett. A, 279 (2001), 333 | DOI | MR | Zbl

[19] A. Minic, Phys. Lett. B, 536 (2002), 305 | DOI | MR | Zbl

[20] S. V. Manakov, Funkts. analiz i ego prilozh., 10 (1976), 93 ; M. Adler, P. van Moerbeke, Adv. Math., 38 (1980), 267 | MR | Zbl | DOI | Zbl

[21] J. Cieslinski, M. Czachor, N. Ustinov, J. Math. Phys., 44 (2003), 1763 | DOI | MR | Zbl

[22] S. Leble, “Covariant forms of Lax one-field operators: from Abelian to non-commutative”, Bilinear Integrable Systems: from Classical to Quantum, from Continuous to Discrete, Proc. NATO Advanced Research Workshop (St. Petersburg, Russia, 15–19 September, 2002), NATO Sci. Ser. II: Math. Phys. Chem., 201, eds. L. Faddeev, P. van Moerbeke, F. Lambert, Kluwer, Dordrecht, 2005 ; E-print math-ph/0302053 | MR

[23] S. Leble, “Dressing chain equations associated with difference soliton systems”, Probing the structure of quantum mechanics. Nonlinearity, nonlocality, computation, axiomatics, eds. D. Aerts, M. Czachor, T. Durt, World Scientific, London, 2002, 354 | DOI | MR | Zbl

[24] J. Weiss, J. Math. Phys., 27 (1986), 2647 | DOI | MR | Zbl

[25] A. P. Veselov, A. B. Shabat, Funkts. analiz i ego prilozh., 27 (1993), 1 | MR | Zbl

[26] V. E. Adler, V. G. Marikhin, A. B. Shabat, TMF, 129:2 (2001), 163 | DOI | MR | Zbl

[27] A. B. Shabat, TMF, 121:1 (1999), 165 | DOI | MR | Zbl

[28] W. Nahm, Phys. Lett. B, 90 (1980), 413 ; N. J. Hitchin, Commun. Math. Phys., 89:2 (1983), 145 | DOI | MR | DOI | MR | Zbl

[29] R. W. Atherton, G. M. Homsy, Studies Appl. Math., 54 (1975), 31 | DOI | MR | Zbl