Liu Shi-Xing

Nonlinear resonance phenomenon of one-dimensional Bose—Einstein condensate under periodic modulation

We investigate the effect of an external periodic modulation on the one-dimensional (1D) Bose—Einstein condensate with harmonic trapping potential. By numerically solving the Gross—Pitaevskii equation with symplectic algorithm, the nonlinear resonance phenomenon is shown and the corresponding Fourier spectrum is given. The autoresonance phenomenon is also presented under almost periodic external modulation, and it shows that the condensate eventually evolves into quasi-periodic oscillation.

Research on the discrete variational method for a Birkhoffian system

In this paper, we present a new integration algorithm based on the discrete Pfaff—Birkhoff principle for Birkhoffian systems. It is proved that the new algorithm can preserve the general symplectic geometric structures of Birkhoffian systems. A numerical experiment for a damping oscillator system is conducted. The result shows that the new algorithm can better simulate the energy dissipation than the R-K method, which illustrates that we can numerically solve the dynamical equations by the discrete variational method in a Birkhoffian framework for the systems with a general symplectic structure. Furthermore, it is demonstrated that the results of the numerical experiments are determined not by the constructing methods of Birkhoffian functions but by whether the numerical method can preserve the inherent nature of the dynamical system.

A necessary and sufficient condition for transforming autonomous systems into linear autonomous Birkhoffian systems

The problem of transforming autonomous systems into Birkhoffian systems is studied. A reasonable form of linear autonomous Birkhoff equations is given. By combining them with the undetermined tensor method, a necessary and sufficient condition for an autonomous system to have a representation in terms of linear autonomous Birkhoff equations is obtained. The methods of constructing Birkhoffian dynamical functions are given. Two examples are given to illustrate the application of the results.

Birkhoffian representation of non-homogenous hamiltonian systems

The relation between Hamiltonian system and Birkhoffian system is discussed in this article. Simultaneously, the theory significance and the practical value of Birkhoffian dynamical systems are also investigated. Furthermore, the Birkhoffian realization theory and methods for constructing Birkhoff’s equation are also studied. Then the primary difficulty and the important investigative directions of Birkhoffian dynamics are pointed out in this article. Finally, the formulations and the significance of generalized Birkhoffian dynamics are given. At the same time the almost-generalized Birkhoff’s equations and its applications are also discussed in briefly.

Decomposition of almost-Poisson structure of generalised Chaplygin's nonholonomic systems

This paper constructs an almost-Poisson structure for the non-self-adjoint dynamical systems, which can be decomposed into a sum of a Poisson bracket and the other almost-Poisson bracket. The necessary and sufficient condition for the decomposition of the almost-Poisson bracket to be two Poisson ones is obtained. As an application, the almost-Poisson structure for generalised Chaplygins systems is discussed in the framework of the decomposition theory. It proves that the almost-Poisson bracket for the systems can be decomposed into the sum of a canonical Poisson bracket and another two noncanonical Poisson brackets in some special cases, which is useful for integrating the equations of motion.

Conformal invariance and Hojman conserved quantities of canonical Hamilton systems

This paper discusses the conformal invariance by infinitesimal transformations of canonical Hamilton systems. The necessary and sufficient conditions of conformal invariance being Lie symmetrical simultaneously by the action of infinitesimal transformations are given. The determining equations of the conformal invariance are gained. Then the Hojman conserved quantities of conformal invariance by special infinitesimal transformations are obtained. Finally an illustrative example is given to verify the results.