Structure of Binary Transformations of Darboux Type and Their Application to Soliton Theory
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On the basis of generalized Lagrange identity for pairs of formally adjoint multidimensional differential operators and a special differential geometric structure associated with this identity, we propose a general scheme of the construction of corresponding transformation operators that are described by nontrivial topological characteristics. We construct explicitly the corresponding integro-differential symbols of transformation operators, which are used in the construction of Lax-integrable nonlinear two-dimensional evolutionary equations and their Darboux–Bäcklund-type transformations.
KeywordsEvolutionary Equation Differential Operator General Scheme Geometric Structure Topological Characteristic
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- 1.V. B. Matveev and M. I. Salle, Darboux Transformations and Solutions, Springer, Berlin (1991).Google Scholar
- 2.J. C. C. Nimmo, Darboux Transformations from Reductions of the KP-Hierarchy, Preprint, November 8, 2002, Glasgow University, Glasgow (2002).Google Scholar
- 3.A. M. Samoilenko and Ya. A. Prykarpats’kyi, Algebro-Analytic Aspects of Completely Integrable Dynamical Systems and Their Perturbations [in Ukrainian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (2002).Google Scholar
- 4.S. P. Novikov (editor), Soliton Theory: Method of Inverse Problem [in Russian], Nauka, Moscow (1980).Google Scholar
- 5.L. D. Faddeev, “The inverse problem of the quantum scattering theory,” in: VINITI Series in Contemporary Problems of Mathematics [in Russian], VINITI, Moscow (1974), pp. 93-180.Google Scholar
- 6.L. P. Nizhnik, “Integration of many-dimensional nonlinear equations by the method of inverse problem,” Dokl. Akad. Nauk SSSR, 254, No. 2, 332-335 (1980).Google Scholar
- 7.L. P. Nizhnik, Inverse Scattering Problems for Hyperbolic Equations [in Russian], Naukova Dumka, Kiev (1991).Google Scholar
- 8.S. V. Manakov, “Method of inverse scattering transform and two-dimensional evolution problems,” Usp. Mat. Nauk, 31, No. 5, 245-246 (1976).Google Scholar
- 9.C. Godbillon, Geometrie Differentielle et Mechanique Analytique, Hermann, Paris (1969).Google Scholar
- 10.H. Cartan, Calcul Differentiel. Formes Differentielles, Hermann, Paris (1967).Google Scholar
- 11.R. Hirota and J. Satsuma, “On an integrable equations related with matrix differential operators,” Phys. Lett., 85, 407-412 (1981).Google Scholar