Yufu Chen

Symbolic computation of Jacobi elliptic function solutions to nonlinear differential-difference equations

In this paper, an algorithm is presented to find exact polynomial solutions of nonlinear differential-difference equations(DDEs) in terms of the Jacobi elliptic functions. The key steps of the algorithm are illustrated by the discretized mKdV lattice. A Maple package JACOBI is developed based on the algorithm to automatically compute special solutions of nonlinear DDEs. The effectiveness of the package is demonstrated by applying it to a variety of equations.

Full Quantum Treatment of Rabi Oscillation Driven by a Pulse Train and Its Application in Ion-Trap Quantum Computation

Rabi oscillation of a two-level system driven by a pulse train is a basic process involved in quantum computation. We present a full quantum treatment of this process and show that the population inversion of this process collapses exponentially, has no revival phenomenon, and has a dual-pulse structure in every period. As an application, we investigate the properties of this process in ion-trap quantum computation. We find that in the Cirac-Zoller computation scheme, when the wavelength of the driving field is of the order 10-6 m , the lower bound of failure probability is of the order 10-2 after about 102 controlled-NOT gates. This value is approximately equal to the generally-accepted threshold in fault-tolerant quantum computation.

Classification and approximate solutions to perturbed diffusion–convection equations

Abstract Approximate symmetries of the perturbed nonlinear diffusion–convection equations are completely classified by the method originated with Fushchich and Shtelen. Moreover, for some interesting cases, symmetry reductions and approximate solutions are discussed in detail.

Comparison of approximate symmetry and approximate homotopy symmetry to the Cahn-Hilliard equation

Complete infinite order approximate symmetry and approximate homotopy symmetry classifications of the Cahn-Hilliard equation are performed and the reductions are constructed by an optimal system of one-dimensional subalgebras. Zero order similarity reduced equations are nonlinear ordinary differential equations while higher order similarity solutions can be obtained by solving linear variable coefficient ordinary differential equations. The relationship between two methods for different order are studied and the results show that the approximate homotopy symmetry method is more effective to control the convergence of series solutions than the approximate symmetry one.

Mixed moving finite element method

Abstract In this paper, we present a mixed moving finite element method (MMFEM). In this method, nodes are divided into several classes and move differently. We concentrate on Burgers’ equation with a linear basis function and carry out the element and global analysis. The computation shows the improvement in contrast to moving finite element method.

A Decoherent Limit of Fault-Tolerant Quantum Computation Driven by Coherent Fields

Based on amplitude character of quantum Rabi oscillation driven by coherent field we show that there exists an upper bound to logic complexity of quantum circuit. We introduce a parameter called single-qubit logic complexity and estimate its decoherence limit in a reasonable case. The analysis show that a generally accepted constant threshold of the threshold theorem limits the logic complexity to so small a number that even a typical construction of fault-tolerant quantum Toffoli gate can hardly be implemented reliably. This result suggests that the construction of feasible fault-tolerant quantum gates is still an arduous task.

Direct reductions of a nonlinear partial differential system

Based upon the Clarkson and Kruskal direct method, we investigate an integrable system using the symbolic computation system. Several types of similarity reductions are obtained.

NONLOCAL SYMMETRIES AND NONLOCAL RECURSION OPERATORS

Abstract An expose about covering method on differential equations was given. The general formulae to determine nonlocal symmetries were derived which are analogous to the prolongation formulae of generalized symmetries. In addition, a new definition of nonlocal recursion operators was proposed, which gave a satisfactory explaination in covering theory for the integro-differential recursion operators.