Том 189, № 1 (2016)

Considering the recent result that the Poisson–Nijenhuis geometry corresponds to the quantization of the symplectic groupoid integrating a Poisson manifold, we discuss the Poisson–Nijenhuis structure on the Grassmannian defined by the compatible Kirillov–Kostant–Souriau and Bruhat–Poisson structures. The eigenvalues of the Nijenhuis tensor are Gelfand–Tsetlin variables, which, as was proved, are also in involution with respect to the Bruhat–Poisson structure. Moreover, we show that the Stiefel bundle on the Grassmannian admits a bi-Hamiltonian structure.

We study the chaos control and the function projective synchronization of a fractional-order T-system and Lorenz chaotic system using the backstepping method. Based on stability theory, we consider the condition for the local stability of nonlinear three-dimensional commensurate fractional-order system. Using the feedback control method, we control the chaos in the considered fractional-order T-system. We simulate the function projective synchronization between the fractional-order T-system and Lorenz system numerically using MATLAB and depict the results with plots.

Written in the evolutionary form, the multidimensional integrable dispersionless equations, exactly like the soliton equations in 2+1 dimensions, become nonlocal. In particular, the Pavlov equation is brought to the form v_t = v_xv_y - ∂_x^-1∂_y[v_y + v_x²], where the formal integral ∂_x⁻¹ becomes the asymmetric integral \( - \int_x^\infty {dx'} \) . We show that this result could be guessed using an apparently new integral geometry lemma. It states that the integral of a sufficiently general smooth function f(X, Y) over a parabola in the plane (X, Y) can be expressed in terms of the integrals of f(X, Y) over straight lines not intersecting the parabola. We expect that this result can have applications in two-dimensional linear tomography problems with an opaque parabolic obstacle.

We survey the matrix product solutions of the Yang–Baxter equation recently obtained from the tetrahedron equation. They form a family of quantum R-matrices of generalized quantum groups interpolating the symmetric tensor representations of U_q(A_n−1⁽¹⁾) and the antisymmetric tensor representations of \({U_{ - {q^{ - 1}}}}\left( {A_{n - 1}^{\left( 1 \right)}} \right)\) . We show that at q = 0, they all reduce to the Yang–Baxter maps called combinatorial R-matrices and describe the latter by an explicit algorithm.

In the framework of approximate symmetries, we investigate a perturbed system of partial differential equations for viscoelastic media with nonlinear dissipation. We completely classify the approximate symmetries and prove a theorem on the relation between the symmetries of two related models. In some physical cases, we find approximate solutions using the generator of the group of transformations taken in the first-order approximation.