Search

We present a machine learning based method for learning first integrals of systems of ordinary differential equations from given trajectory data. The method is model-agnostic in that it does not require explicit knowledge of the underlying system of differential equations that generated the trajectories. As a by-product, once the first integrals have been learned, also the system of differential equations will be known. We illustrate our method by considering several classical problems from the mathematical sciences., Comment: 18 pages, 7 figures

In the present paper we consider nonlinear multidimensional Cahn-Hilliard and Kuramoto-Sivashinsky equations that have many important applications in physics and chemistry, and a certain natural generalization of these equations. For an arbitrary number of spatial independent variables we present a complete list of cases when the generalized Cahn-Hilliard-Kuramoto-Sivashinsky equation admits nontrivial local conservation laws of any order, and for each of those cases we give an explicit form of all the local conservation laws of all orders modulo trivial ones admitted by the equation under study. In particular, we show that the original Kuramoto-Sivashinsky equation admits no nontrivial local conservation laws, and find all nontrivial local conservation laws for the Cahn-Hilliard equation., Comment: 10 pages, no figures

We prove that the generalized Swift--Hohenberg equation with nonlinear right-hand side, a natural generalization of the Swift--Hohenberg equation arising in physics and describing inter alia pattern formation, has no nontrivial local conservation laws., Comment: 3 pages, no figures

We consider the equations $u_{yy} = u_y u_{xx} - (u_x + u)u_{xy} + u_x u_y$ and $u_{yy} = (u_x + x)u_{xy} - (u_{xx} + 2)u_y$ that arise as reductions of the universal hierarchy and rdDym equations, respectively, and describe the Lie algebras of nonlocal symmetries in infinite-dimensional coverings naturally associated to these equations.

We consider the 3D equation $u_{yy} = u_{tx} + u_yu_{xx} - u_xu_{xy}$ and its 2D reductions: (1) $u_{yy} = (u_y+y)u_{xx}-u_xu_{xy}-2$ (which is equivalent to the Gibbons-Tsarev equation) and (2) $u_{yy} = (u_y+2x)u_{xx} + (y-u_x)u_{xy} -u_x$. Using reduction of the known Lax pair for the 3D equation, we describe nonlocal symmetries of~(1) and~(2) and show that the Lie algebras of these symmetries are isomorphic to the Witt algebra.

With the advent of the wide-spread use of smart phones, video streaming over mobile wireless networks has suddenly taken a huge surge in recent years. Considering its enormous potential, mobile WiMAX is emerging as a viable technology for mobile TV which is expected to become of key importance in the future of mobile indus- try. In this paper, a simulation performance study of Mobile TV over mobile WiMAX is conducted with different types of adaptive modulation and coding taking into account key system and environment parameters which include the variation in the speed of the mobile, path-loss, scheduling service classes with the fixed type of mod- ulations. Our simulation has been conducted using OPNET simulation. Simulation results show that dynamic adaptation of modulation and coding schemes based onchannel conditions can offer considerably more en- hanced QoS and at the same time reduce the overall bandwidthof the system., Comment: 12 pages, 9 figures. arXiv admin note: substantial text overlap with arXiv:1312.7442; and text overlap with arXiv:1005.0976 by other authors