Turning points and traveling waves in FitzHugh-Nagumo type equations

Consider the following FitzHugh-Nagumo type equation

     

WAVES IN FERROMAGNETIC MEDIA

ABSTRACT

It is shown that small perturbations of equilibrium states in ferromagnetic media give rise to standing and traveling waves that are stable for long times. The evolution of the wave profiles is governed by semilinear heat equations. The mathematical model underlying these results consists of the Landau–Lifshitz equation for the magnetization vector and Maxwell's equations for the electromagnetic field variables. The model belongs to a general class of hyperbolic equations for vector-valued functions, whose asymptotic properties are analyzed rigorously. The results are illustrated with numerical examples.     

Packing circuits in matroids

The purpose of this paper is to characterize all matroids M that satisfy the following minimax relation: for any nonnegative integral weight function w defined on E(M),

The Approach of Solutions for the Nonlocal Diffusion Equation to Traveling Fronts

The paper is concerned with the asymptotic behavior as t → ±∞ of an entire solution u(x,t) for the nonlocal diffusion equation. With bistable assumption, it is well known that the model has three different types of travel- ing fronts. Under certain conditions on the wave speeds, and by some auxiliary rational functions with certain properties to construct appropriate super- and sub solutions of the model, we establish two new types of entire solutions u(x,t) which approach to three travelling fronts or the positive equilibrium as t → ±∞.     

A 4/3-approximation for TSP on cubic 3-edge-connected graphs

For a 3-edge-connected cubic graph $G=(V,E)$, we give an algorithm to construct a connected Eulerian subgraph of $2G$ using at most $?4\left|V\right|∕3?$ edges.     

A General VNS heuristic for the traveling salesman problem with time windows

This paper presents a General Variable Neighborhood Search (GVNS) heuristic for the Traveling Salesman Problem with Time Windows (TSPTW). The heuristic is composed by both constructive and optimization stages. In the first stage, the heuristic constructs a feasible solution using VNS, and in the optimization stage the heuristic improves the feasible solution with a General VNS heuristic. Both constructive and optimization stages take advantage of elimination tests, partial neighbor evaluation and neighborhood partitioning techniques. Experimental results show that this approach is efficient, reducing significantly the computation time and improving some best known results from the literature.     

Traveling wave solutions in delayed nonlocal diffusion systems with mixed monotonicity

This paper deals with the existence of traveling wave solutions in delayed nonlocal diffusion systems with mixed monotonicity. Based on two different mixed-quasimonotonicity reaction terms, we propose new definitions of upper and lower solutions. By using Schauder's fixed point theorem and a new cross-iteration scheme, we reduce the existence of traveling wave solutions to the existence of a pair of upper and lower solutions. The general results obtained have been applied to type-K monotone and type-K competitive nonlocal diffusive Lotka-Volterra systems.     

The traveling salesman problem with pickup and delivery: polyhedral results and a branch-and-cut algorithm

The Traveling Salesman Problem with Pickup and Delivery (TSPPD) is defined on a graph containing pickup and delivery vertices between which there exists a one-to-one relationship. The problem consists of determining a minimum cost tour such that each pickup vertex is visited before its corresponding delivery vertex. In this paper, the TSPPD is modeled as an integer linear program and its polyhedral structure is analyzed. In particular, the dimension of the TSPPD polytope is determined and several valid inequalities, some of which are facet defining, are introduced. Separation procedures and a branch-and-cut algorithm are developed. Computational results show that the algorithm is capable of solving to optimality instances involving up to 35 pickup and delivery requests, thus more than doubling the previous record of 15.      

The exponential function rational expansion method and exact solutions to nonlinear lattice equations system

We propose an exponential function rational expansion method for solving exact traveling wave solutions to nonlinear differential-difference equations system. By this method, we obtain some exact traveling wave solutions to the relativistic Toda lattice equations system and discuss the significance of these solutions. Finally, we give an open problem.     

A sub-ODE method for generalized Gardner and BBM equation with nonlinear terms of any order

In this paper, a method with the aid of a sub-ODE and its solutions is used for constructing new periodic wave solutions for nonlinear Gardner equation and BBM equation with nonlinear terms of any order arising in mathematical physics. As a result, many exact traveling wave solutions are successfully obtained. The method in the paper is very direct and it can also be applied to other nonlinear evolution equations.