Some Exact Solutions of Two Fifth Order KdV-Type Nonlinear Partial Differential Equations

We consider the generalized integrable fifth order nonlinear Korteweg-de Vries （fKdV） equation. The extended Tanh method has been used rigorously, by com- putational program MAPLE, for solving this fifth order nonlinear partial differential equation. The general solutions of the fKdV equation are formed considering an ansatz of the solution in terms of tanh. Then, in particular, some exact solutions are found for the two fifth order KdV-type equations given by the Caudrey-Dodd-Gibbon equation and the another fifth order equation. The obtained solutions include solitary wave solution for both the two equations.

Orbital dynamics satisfying the 4^(th)-order stationary extended Fisher-Kolmogorov equation

In this study,we discuss the central force problem by using the nonlocal-in-time kinetic energy approach.At low length scales,the system is dominated by the generalized 4^(th)-order extended Fisher-Kolmogorov stationary equation and by the 4^(th)-order stationary Swift-Hohenberg differential equation under explicit conditions.The energy is a conserved quantity along orbits of the extended Fisher-Kolmogorov stationary equation.The system is quantized,the system is stable,and the ground energy problem is solved.

Pattern control for large-scale spacecraft swarms in elliptic orbits via density fields

Space swarms,enabled by the miniaturization of spacecraft,have the potential capability to lower costs,increase efficiencies,and broaden the horizons of space missions.The formation control problem of large-scale spacecraft swarms flying around an elliptic orbit is considered.The objective is to drive the entire formation to produce a specified spatial pattern.The relative motion between agents becomes complicated as the number of agents increases.Hence,a density-based method is adopted,which concerns the density evolution of the entire swarm instead of the trajectories of individuals.The density-based method manipulates the density evolution with Partial Differential Equations(PDEs).This density-based control in this work has two aspects,global pattern control of the whole swarm and local collision-avoidance between nearby agents.The global behavior of the swarm is driven via designing velocity fields.For each spacecraft,the Q-guidance steering law is adopted to track the desired velocity with accelerations in a distributed manner.However,the final stable velocity field is required to be zero in the classical density-based approach,which appears as an obstacle from the viewpoint of astrodynamics since the periodic relative motion is always time-varying.To solve this issue,a novel transformation is constructed based on the periodic solutions of Tschauner-Hempel(TH)equations.The relative motion in Cartesian coordinates is then transformed into a new coordinate system,which permits zero-velocity in a stable configuration.The local behavior of the swarm,such as achieving collision avoidance,is achieved via a carefully-designed local density estimation algorithm.Numerical simulations are provided to demonstrate the performance of this approach.