A new (abstract algebraic) approach to the solution of the order conditions for Runge-Kutta methods (RK) and to the corresponding simplifying assumptions was suggested in [9, 10]. The approach implied natural classification of the simplifying assumptions and allowed to find new RK methods of high orders. Here we further this approach. The new approach is based on the upper and lower Butcher’s a...

We construct a tau cover of the generalized Drinfeld–Sokolov hierarchy associated with an arbitrary affine Kac–Moody algebra gradations s ? 1 $\mathrm{s}\leqslant \mathbb {1}$ and derive its Virasoro symmetries. By imposing constraints we obtain solutions Witten–Kontsevich Brezin–Gross–Witten types, those characterized by certain ordinary differential equations Painlevé type. also show existenc...

The associativity equations, also known as the Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations [1, 2], and the related geometric structures, namely, the Frobenius manifolds [3, 4, 5, 6, 7, 8], have recently attracted considerable attention because of their manifold applications in physics and mathematics. More recently, the oriented associativity equations, a generalization of the WDVV equa...

A number of interesting problems can be modeled with nonlinear differential-difference equations (DDEs) [1][3], including particle vibrations in lattices, currents in electrical networks, and pulses in biological chains. Nonlinear DDEs also play a role in queuing problems and discretizations in solid state and quantum physics, and arise in the numerical solution of nonlinear PDEs. The study of ...

This paper addresses the problem of stabilization non-minimum phase switched nonlinear systems where internal dynamics with symmetries or non-symmetries each mode may be unstable. The authors initially build a stabilizing Lyapunov controller for in order to stabilize its own unstable dynamics. proposed approach is based on exact input-output feedback linearization technique and stability theory...

The developments in this paper are concerned with nonholonomic field theories in the presence of symmetries. Having previously treated the case of vertical symmetries, we now deal with the case where the symmetry action can also have a horizontal component. As a first step in this direction, we derive a new and convenient form of the field equations of a nonholonomic field theory. Nonholonomic ...

Given a generalized (Lie–Bäcklund) vector field, we classify (under certain nondegeneracy assumptions) all (1+1)-dimensional evolution systems which admit this vector field as a conditional symmetry. The connection with the theory of symmetries of systems of ODEs and with the theory of invariant modules is discussed.