The homogeneous balance method and its application to the Benjamin-Bona-Mahoney (BBM) equation

We make use of the homogeneous balance method and symbolic computation to construct new exact traveling wave solutions for the Benjamin-Bona-Mahoney (BBM) equation. Many new exact traveling wave solutions are successfully obtained, which contain rational and periodic-like solutions. This method is straightforward and concise, and it can also be applied to other nonlinear evolution equations.     

Forced waves of the Fisher–KPP equation in a shifting environment

This paper concerns the equation

(0.1)$u_t=u_{xx}+f(x?ct,u),x\in \mathbb{R},$

where $c\ge 0$ is a forcing speed and $f:(s,u)\in \mathbb{R}\times {\mathbb{R}}_{+}\to \mathbb{R}$ is asymptotically of KPP type as $s\to ?\infty$. We are interested in the questions of whether such a forced moving KPP nonlinearity from behind can give rise to traveling waves with the same speed and how they attract solutions of initial value problems when they exist. Under a sublinearity condition on $f(s,u)$, we obtain the complete existence and multiplicity of forced traveling waves as well as their attractivity except for some critical cases. In these cases, we provide examples to show that there is no definite answer unless one imposes further conditions depending on the heterogeneity of f in $s\in \mathbb{R}$.     

A hyperbolic model for the effects of urbanization on air pollution

A hyperbolic model to study effects of industrialization and urbanization on air pollution propagation is proposed.     

Validating vehicle routing zone construction using Monte Carlo simulation

The primary purpose of this paper is to validate a clustering procedure used to construct contiguous vehicle routing zones (VRZs) in metropolitan regions. Given a set of customers with random demand for pickups and deliveries over the day, the goal of the design problem is to cluster the customers into zones that can be serviced by a single vehicle. Monte Carlo simulation is used to determine the feasibility of the zones with respect to package count and tour time. For each replication, a separate probabilistic traveling salesman problem (TSP) is solved for each zone. For the case where deliveries must precede pickups, a heuristic approach to the TSP is developed and evaluated, also using Monte Carlo simulation. In the testing, performance is measured by overall travel costs and the probability of constraint violations. Gaps in tour length, tour time and tour cost are the measure used when comparing exact and heuristic TSP solutions.     

A new branch-and-price algorithm for the traveling tournament problem

The traveling tournament problem (ttp) consists of finding a distance-minimal double round-robin tournament where the number of consecutive breaks is bounded. For solving the problem exactly, we propose a new branch-and-price approach. The starting point is a new compact formulation for the ttp. The corresponding extensive formulation resulting from a Dantzig-Wolfe decomposition is identical to one given by Easton, K., Nemhauser, G., Trick, M., 2003. Solving the traveling tournament problem: a combined interger programming and constraint programming approach. In: Burke, E., De Causmaecker, P. (Eds.), Practice and Theory of Automated Timetabling IV, Volume 2740 of Lecture Notes in Computer Science, Springer Verlag Berlin/Heidelberg, pp. 100–109, who suggest to solve the tour-generation subproblem by constraint programming. In contrast to their approach, our method explicitly utilizes the network structure of the compact formulation: First, the column-generation subproblem is a shortest-path problem with additional resource and task-elementarity constraints. We show that this problem can be reformulated as an ordinary shortest-path problem over an expanded network and, thus, be solved much faster. An exact variable elimination procedure then allows the reduction of the expanded networks while still guaranteeing optimality. Second, the compact formulation gives rise to supplemental branching rules, which are needed, since existing rules do not ensure integrality in all cases. Third, non-repeater constraints are added dynamically to the master problem only when violated. The result is a fast exact algorithm, which improves many lower bounds of knowingly hard ttp instances from the literature. For some instances, solutions are proven optimal for the first time.     

The Attractive Traveling Salesman Problem

In the Attractive Traveling Salesman Problem the vertex set is partitioned into facility vertices and customer vertices. A maximum profit tour must be constructed on a subset of the facility vertices. Profit is computed through an attraction function: every visited facility vertex attracts a portion of the profit from the customer vertices based on the distance between the facility and customer vertices, and the attractiveness of the facility vertex. A gravity model is used for computing the profit attraction. The problem is formulated as an integer non-linear program. A linearization is proposed and strengthened through the introduction of valid inequalities, and a branch-and-cut algorithm is developed. A tabu search algorithm is also implemented. Computational results are reported.     

Research on traveling wave solutions for a class of (3+1)-dimensional nonlinear equation

Nonlinear wave phenomena are of great importance in the nature, and have became for a long time a challenging research topic for both pure and applied mathematicians. In this paper the solitary wave, kink (anti-kink) wave and periodic wave solutions for a class of (3+1)-dimensional nonlinear equation were obtained by some effective methods from the dynamical systems theory.     

Minimum wave speed for a diffusive competition model with time delay

In this paper we study mono-stable traveling wave solutions for a Lotka-Volterra reaction-diffusion competition model with time delay. By constructing upper and lower solutions, we obtain the precise minimum wave speed of traveling waves under certain conditions. Our results also extend the known results on the minimum wave speed for Lotka-Volterra competition model without delay.     

Stability and traveling waves of an epidemic model with relapse and spatial diffusion

An epidemic model with relapse and spatial diffusion is studied. Such a model is appropriate for tuberculosis, including bovine tuberculosis in cattle and wildlife, and for herpes. By using the linearized method, the local stability of each of feasible steady states to this model is investigated. It is proven that if the basic reproduction number is less than unity, the disease-free steady state is locally asymptotically stable; and if the basic reproduction number is greater than unity, the endemic steady state is locally asymptotically stable. By the cross-iteration scheme companied with a pair of upper and lower solutions and Schauder's fixed point theorem, the existence of a traveling wave solution which connects the two steady states is established. Furthermore, numerical simulations are carried out to complement the main results.