Generalized Camassa–Holm Equations: Symmetry, Conservation Laws and Regular Pulse and Front Solutions

In this paper, we consider a member of an integrable family of generalized Camassa–Holm (GCH) equations. We make an analysis of the point Lie symmetries of these equations by using the Lie method of infinitesimals. We derive nonclassical symmetries and we find new symmetries via the nonclassical method, which cannot be obtained by Lie symmetry method. We employ the multiplier method to construct conservation laws for this family of GCH equations. Using the conservation laws of the underlying equation, double reduction is also constructed. Finally, we investigate traveling waves of the GCH equations. We derive convergent series solutions both for the homoclinic and heteroclinic orbits of the traveling-wave equations, which correspond to pulse and front solutions of the original GCH equations, respectively.

The Effect of Different Decision-Making Methods on Multi-Objective Optimisation of Predictive Torque Control Strategy

Today, a clear trend in electrification process has emerged in all areas to cope with carbon emissions. For this purpose, the widespread use of electric cars and wind energy conversion systems has increased the attention and importance of electric machines. To overcome limitations in mature control techniques, model predictive control (MPC) strategies have been proposed. Of these strategies, predictive torque control (PTC) has been well accepted in the control of electric machines. However, it suffers from the selection of weighting factors in the cost function. In this paper, the weighting factor associated with the flux error term is optimised by the non-dominated sorting genetic algorithm (NSGA-II) algorithm through torque and flux errors. The NSGA-II algorithm generates a set of optimal solutions called Pareto front solutions, and a possible solution must be selected from among the Pareto front solutions for use in the PTC strategy. Unlike the current literature, three decision-making methods are applied to the Pareto front solutions and the weighting factors selected by each method are tested under different operating conditions in terms of torque ripples, flux ripples, cur-rent harmonics and average switching frequencies. Finally, a decision-making method is recommended.

Bi-Objective Dynamic Multiprocessor Open Shop Scheduling: An Exact Algorithm

An important element in the integration of the fourth industrial revolution is the development of efficient algorithms to deal with dynamic scheduling problems. In dynamic scheduling, jobs can be admitted during the execution of a given schedule, which necessitates appropriately planned rescheduling decisions for maintaining a high level of performance. In this paper, a dynamic case of the multiprocessor open shop scheduling problem is addressed. This problem appears in different contexts, particularly those involving diagnostic operations in maintenance and health care industries. Two objectives are considered simultaneously—the minimization of the makespan and the minimization of the mean weighted flow time. The former objective aims to sustain efficient utilization of the available resources, while the latter objective helps in maintaining a high customer satisfaction level. An exact algorithm is presented for generating optimal Pareto front solutions. Despite the fact that the studied problem is NP-hard for both objectives, the presented algorithm can be used to solve small instances. This is demonstrated through computational experiments on a testbed of 30 randomly generated instances. The presented algorithm can also be used to generate approximate Pareto front solutions in case computational time needed to find proven optimal solutions for generated sub-problems is found to be excessive. Furthermore, computational results are used to investigate the characteristics of the optimal Pareto front of the studied problem. Accordingly, some insights for future metaheuristic developments are drawn.

Entire solutions for a delayed lattice competitive system

In this paper, we investigate the existence of entire solutions for a delayed lattice competitive system. Here the entire solutions are the solutions that exist for all $(n,t)\in \mathbb{Z}\times \mathbb{R}$ ( n , t ) ∈ Z × R . In order to prove the existence, we firstly embed the delayed lattice system into the corresponding larger system, of which the traveling front solutions are identical to those of the delayed lattice system. Then based on the comparison theorem and the sup–sub solutions method, we construct entire solutions which behave as two opposite traveling front solutions moving towards each other from both sides of x-axis and then annihilating. Moreover, our conclusions extend the invading way, which the superior species invade the inferior ones from both sides of x-axis and then the inferior ones extinct, into the lattice and delay case.