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For Hermitian adjoint differential operators, we consider the structure of Darboux–Bäcklund-type transformations in the class of parametrically dependent Hilbert spaces. By using the proposed new method, we obtain the corresponding integro-differential symbols of the operators of transformations in explicit form and consider the problem of their application to the construction of two-dimensional Lax-integrable nonlinear evolution equations and their Darboux–Bäcklund-type transformations. 相似文献
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Samoilenko A. M. Prykarpats'kyi A. K. Samoilenko V. H. 《Ukrainian Mathematical Journal》2003,55(1):82-92
We analyze the geometric structure of the Lyapunov–Schmidt approach to studying critical manifolds of weakly perturbed Lagrangian and Hamiltonian systems. 相似文献
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We study the problem of the complete integrability of nonlinear oscillatory dynamical systems connected, in particular, both with the Cartan decomposition of a Lie algebra
is the Lie algebra of a fixed subgroup
with respect to an involution : G G on the Lie group G, and with a Poisson action of special type on a symplectic matrix manifold. 相似文献
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A class of dynamical systems on symplectic manifolds solving linear programming problems is described. The structure of an orbit space is analyzed within the framework of the Marsden–Weinstein reduction scheme. Some examples having applications in modern macroeconomic modeling are studied in detail. 相似文献
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Prykarpats'kyi Ya. A. Samoilenko A. M. Samoilenko V. H. 《Ukrainian Mathematical Journal》2003,55(12):2041-2059
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. 相似文献
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Prykarpats'kyi A. K. Brzychczy S. Samoilenko V. G. 《Ukrainian Mathematical Journal》2001,53(2):249-258
We study infinite-dimensional Liouville–Lax integrable nonlinear dynamical systems. For these systems, we consider the problem of finding an appropriate set of initial conditions leading to typical solutions such as solitons and traveling waves. We develop an approach to the solution of this problem based on the exact reduction of a given nonlinear dynamical system to its finite-dimensional invariant submanifolds and the subsequent investigation of the system of ordinary differential equations obtained by qualitative analysis. The efficiency of the approach proposed is demonstrated by the examples of the Korteweg–de Vries equation, the modified nonlinear Schrödinger equation, and a hydrodynamic model. 相似文献
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