In this present study analytical method based on Riccati Equation as for converting the Nonlinear Lakshmanan-Porsezian-Daniel (LPD) equation into the nonlinear ODE and finding soliton solutions of this sustem discused. Obtaining solutions are new and obtained from wave transformation. The obtained results show that the presented method is effective and appropriate for solving nonlinear differen...

In this paper a control problem for a controlled linear stochastic equation in a Hilbert space and an exponential quadratic cost functional of the state and the control is formulated and solved. The stochastic equation can model a variety of stochastic partial differential equations with the control restricted to the boundary or to discrete points in the domain. The solution method does not req...

We discuss a parallel algorithm for the solution of large-scale generalized algebraic Riccati equations with dimension up to O(10). We survey the numerical algorithms underlying the implementation of the method, in particular, a Newton-type iterative solver for the generalized Riccati equation and an LR-ADI solver for the generalized Lyapunov equation. Experimental results on a cluster of Intel...

Algebraic geometry plays an important role in the theory of linear systems for (at least) three reasons. First, the Laplace transform turns expressions about linear differential systems into expressions involving rational functions. In addition, many of the transformations studied in linear systems theory, like changes of coordinates or feedback, turn out to be the action of algebraic groups on...

A classical problem in matrix analysis, total least squares estimation and model reduction theory is that of finding a best approxiniant of a given matrix by lower rank ones. It is common believe that behind every such least squares problem there is an algebraic Riccati equation. In this paper we consider the task of minimizing the distance function fA(X) = | |A —A'|| on varieties of fixed rank...

Classical chaos is often characterized as exponential divergence of nearby trajectories. In many interesting cases these trajectories can be identified with geodesic curves. We define here the entropy by S = ln χ(x) with χ(x) being the distance between two nearby geodesics. We derive an equation for the entropy, which by transformation to a Riccati-type equation becomes similar to the Jacobi eq...

We study the complex Riccati tensor equation DċG+GCG−R = 0 on a geodesic c on a Riemannian 3-manifold. This non-linear equation appears in the study of Gaussian beams. Gaussian beams are asymptotic solutions to hyperbolic equations that at each time instant are concentrated around one point in space. When time moves forward, Gaussian beams move along geodesics, and the Riccati equation determin...