A good approximation of modulated amplitude waves in Bose–Einstein condensates

In this paper, we present a perturbation method that utilizes Hamiltonian perturbation theory and averaging to analyze spatio-temporal structures in Gross–Pitaevskii equations and thereby investigate the dynamics of modulated amplitude waves (MAWs) in quasi-one-dimensional Bose–Einstein condensates with mean-field interactions. A good approximation for MAWs is obtained. We also explore dynamics of BECs with the nonresonant external potentials and scatter lengths varying periodically in detail using Hamiltonian perturbation theory and numerical simulations.     

Attitude dynamics of a spacecraft with variable structure at presence of harmonic perturbations

Attitude dynamics of a spacecraft (SC) with variable structure (inertia–mass parameters variation) is examined. Equations of the motion of the SC are obtained on the base of Hamiltonian formalism in Serret–Andoyer variables. These equations can be used for analysis and synthesis of conditions of the SC attitude motion on active legs of orbital trajectories. Analytical and numerical modeling of the SC motion is realized. Existence of the SC chaotic modes of motion is demonstrated with the help of Melnikov method and Poincaré sections. Also attitude motion of a dual-spin spacecraft (DSSC) is considered at presence of small internal harmonic torque between DSSC coaxial bodies.     

On the density of the minimal subspaces generated by discrete linear Hamiltonian systems

This paper focuses on the density of the minimal subspaces generated by a class of discrete linear Hamiltonian systems. It is shown that the minimal subspace is densely defined if and only if the maximal subspace is an operator; that is, it is single valued. In addition, it is shown that, if the interval on which the systems are defined is bounded from below or above, then the minimal subspace is non-densely defined in any non-trivial case.     

Binary Nonlinearization of the Nonlinear Schr?dinger Equation Under an Implicit Symmetry Constraint

By modifying the procedure of binary nonlinearization for the AKNS spectral problem and its adjoint spectral problem under an implicit symmetry constraint,we obtain a finite dimensional system from the Lax pair of the nonlinear Schr¨odinger equation.We show that this system is a completely integrable Hamiltonian system.     

Homoclinic solutions for second‐order discrete Hamiltonian systems with asymptotically quadratic potentials

In this paper, we study the existence of infinitely many homoclinic solutions for the second‐order self‐adjoint discrete Hamiltonian system , where , and are unnecessarily positive definites for all . By using the variant fountain theorem, we obtain an existence criterion to guarantee that the aforementioned system has infinitely many homoclinic solutions under the assumption that W(n,x) is asymptotically quadratic as | x | → + ∞ . Copyright © 2013 John Wiley & Sons, Ltd.     

Variational approach to solutions for a class of fractional Hamiltonian systems

In this paper, we investigate the existence of infinitely many solutions for the following fractional Hamiltonian systems: ( FHS ) where α ∈ (1 ∕ 2,1), , , and are symmetric and positive definite matrices for all , , and ? W is the gradient of W at u. The novelty of this paper is that, assuming L is coercive at infinity, and W is of subquadratic growth as | u | → + ∞ , we show that (FHS) possesses infinitely many solutions via the genus properties in the critical theory. Recent results in the literature are generalized and significantly improved. Copyright © 2013 John Wiley & Sons, Ltd.     

The improved split‐step θ methods for stochastic differential equation

Two improved split‐step θ methods, which, respectively, named split‐step composite θ method and modified split‐step θ‐Milstein method, are proposed for numerically solving stochastic differential equation of Itô type. The stability and convergence of these methods are investigated in the mean‐square sense. Moreover, an approach to improve the numerical stability is illustrated by choices of parameters of these two methods. Some numerical examples show the accordance between the theoretical and numerical results. Further numerical tests exhibit not only the Hamiltonian‐preserving property of the improved split‐step θ methods for a stochastic differential system but also the positivity‐preserving property of the modified split‐step θ‐Milstein method for the Cox–Ingersoll–Ross model. Copyright © 2013 John Wiley & Sons, Ltd.     

A twist condition and multiple solutions of unbounded self-adjoint operator equation with symmetries

By using the index theory for unbounded self-adjoint operator equations and the symmetric mountain pass theorem, we investigate the existence of multiple solutions for nonlinear operator equations with twist conditions. We prove an abstract theorem, and give some applications to first order Hamiltonian systems with Sturm–Liouville boundary conditions and delay differential equations.