Travelling Wave Solutions and Conservation Laws of the (2+1)-dimensional Broer-Kaup-Kupershmidt Equatio

The travelling wave solutions and conservation laws of the (2+1)-dimensional Broer-Kaup-Kupershmidt (BKK) equation are considered in this paper. Under the travelling wave frame, the BKK equation is transformed to a system of ordinary differential equations (ODEs) with two dependent variables. Therefore, it happens that one dependent variable $u$ can be decoupled into a second order ODE that corresponds to a Hamiltonian planar dynamical system involving three arbitrary constants. By using the bifurcation analysis, we obtain the bounded travelling wave solutions $u$, which include the kink, anti-kink and periodic wave solutions. Finally, the conservation laws of the BBK equation are derived by employing the multiplier approach.     

Selective generalized travelling salesman problem

ABSTRACT

This paper introduces the Selective Generalized Traveling Salesman Problem (SGTSP). In SGTSP, the goal is to determine the maximum profitable tour within the given threshold of the tour’s duration, which consists of a subset of clusters and a subset of nodes in each cluster visited on the tour. This problem is a combination of cluster and node selection and determining the shortest path between the selected nodes. We propose eight mixed integer programming (MIP) formulations for SGTSP. All of the given MIP formulations are completely new, which is one of the major novelties of the study. The performance of the proposed formulations is evaluated on a set of test instances by conducting 4608 experimental runs. Overall, 4138 out of 4608 (~90%) test instances were solved optimally by using all formulations.     

Counting and ordering periodic stationary solutions of lattice Nagumo equations

We study the rich structure of periodic stationary solutions of Nagumo reaction diffusion equation on lattices. By exploring the relationship with Nagumo equations on cyclic graphs we are able to divide these periodic solutions into equivalence classes that can be partially ordered and counted. In order to accomplish this, we use combinatorial concepts such as necklaces, bracelets and Lyndon words.     

POPMUSIC for the travelling salesman problem

POPMUSIC— Partial OPtimization Metaheuristic Under Special Intensification Conditions — is a template for tackling large problem instances. This metaheuristic has been shown to be very efficient for various hard combinatorial problems such as p-median, sum of squares clustering, vehicle routing, map labelling and location routing. A key point for treating large Travelling Salesman Problem (TSP) instances is to consider only a subset of edges connecting the cities. The main goal of this article is to present how to build a list of good candidate edges with a complexity lower than quadratic in the context of TSP instances given by a general function. The candidate edges are found with a technique exploiting tour merging and the POPMUSIC metaheuristic. When these candidate edges are provided to a good local search engine, high quality solutions can be found quite efficiently. The method is tested on TSP instances of up to several million cities with different structures (Euclidean uniform, clustered, 2D to 5D, grids, toroidal distances). Numerical results show that solutions of excellent quality can be obtained with an empirical complexity lower than quadratic without exploiting the geometrical properties of the instances.     

Traveling wavefronts for a two-species predator–prey system with diffusion terms and stage structure

In this paper, we considered an important model describing a two-species predator–prey system with diffusion terms and stage structure. By using the linearized method, we investigated the locally asymptotical stability of the nonnegative equilibria of the system and obtained the locally stable conditions. And by using the approach introduced by Canosa [J. Canosa, On a nonlinear diffusion equation describing population growth, IBM J. Res. Dev. 17 (1973) 307–313] and the method of upper and lower solutions, we studied the existence of traveling wavefronts, connecting the zero solution with the positive equilibrium of the system. Our results show that the traveling wavefronts exist and appear to be monotone. Finally, we given a conclusion to summarize the overall achievements of the work presented in the paper.     

Global dynamics of a reaction and diffusion model for an HTLV-I infection with mitotic division of actively infected cells

This paper is concerned with the global dynamics of a reaction and diffusion model for an HTLV-I infection with mitotic division of actively infected cells and CTL immune response. The well posedness of the proposed model is investigated. In the case of a bounded spatial domain, we establish the threshold dynamics in terms of the basic reproduction number $\mathcal{R}_0$ for the spatially heterogeneous model. Also, by means of different Lyapunov functions, the global asymptotic properties of the steady states for the spatially homogeneous model are studied. In the case of an unbounded spatial domain, there are no travelling wave solutions connecting the infection-free steady state with itself when $\mathcal{R}_0 < 1$. Finally, numerical simulations and conclusions are given.     

Interfacial wave propagation due to an interfacial line source

Interfacial wave propagation parallel to a dielectric interface has been studied by considering an electric current line source present at the interface. The first order asymptotic evaluation of field components shows a null of the electric field at the interface. An amplitude null represents an unstable structure in the phase map and a phase front discontinuity across the interface. Higher order asymptotic evaluation has been employed to gain further insight into this propagation problem. The results show that the wavefronts need not be discontinuous. The continuity of the phase fronts is preserved with the help of interesting and stable structures such as saddle points and center points in the phase map of the electric field in both half spaces.      

A rational generalization of Fan’s method and its application to generalized shallow water wave equations

In this paper we employ a rational expansion to generalize Fan’s method for exact travelling wave solutions for nonlinear partial differential equations (PDEs). To verify the reliability of the proposed method, the generalized shallow water wave (GSWW) equation has been investigated as an example. Kinds of new exact travelling wave solutions of a rational form have been obtained. This indicates that the proposed method provides a more general result for exact solution of nonlinear equations.     

Persistence of travelling fronts of KdV-Burgers-Kuramoto equation

We study the travelling fronts of KdV-Burgers-Kuramoto equation from geometric singular perturbation point of view. Motivated by the analogue between travelling fronts and heteroclinic orbits of the corresponding ordinary differential equations, we prove the persistence of these waves for sufficiently small dissipation. The result of numerical investigation also establishes our analysis.     

New exact solutions to the -dimensional Konopelchenko–Dubrovsky equation

In this paper, the extended tanh method, the sech–csch ansatz, the Hirota’s bilinear formalism combined with the simplified Hereman form and the Darboux transformation method are applied to determine the traveling wave solutions and other kinds of exact solutions for the (2+1)-dimensional Konopelchenko–Dubrovsky equation and abundant new soliton solutions, kink solutions, periodic wave solutions and complexiton solutions are formally derived. The work confirms the significant features of the employed methods and shows the variety of the obtained solutions.