Subtour elimination constraints imply a matrix-tree theorem SDP constraint for the TSP

De Klerk et al., (2008) give a semidefinite programming constraint for the Traveling Salesman Problem (TSP) based on the matrix-tree theorem. This constraint says that the aggregate weight of all spanning trees in a solution to a TSP relaxation is at least that of a cycle graph. In this note, we show that the semidefinite constraint holds for any weighted 2-edge-connected graph and, in particular, is implied by the subtour elimination constraints.     

Continuation of homoclinic orbits in the suspension bridge equation: A computer-assisted proof

In this paper, we prove existence of symmetric homoclinic orbits for the suspension bridge equation $u^{?}+\beta u^{″}+e^u?1=0$ for all parameter values $\beta \in [0.5,1.9]$. For each β, a parameterization of the stable manifold is computed and the symmetric homoclinic orbits are obtained by solving a projected boundary value problem using Chebyshev series. The proof is computer-assisted and combines the uniform contraction theorem and the radii polynomial approach, which provides an efficient means of determining a set, centered at a numerical approximation of a solution, on which a Newton-like operator is a contraction.     

Existence and uniqueness of traveling waves for non-monotone integral equations with application

This paper is concerned with the traveling waves in a class of non-monotone integral equations. First we establish the existence of traveling waves. The approach is based on the construction of two associated auxiliary monotone integral equations and a profile set in a suitable Banach space. Then we show that the traveling waves are unique up to translations under some reasonable assumptions. The exact asymptotic behavior of the profiles as ξ→−∞ and the existence of minimal wave speed are also obtained. Finally, we apply our results to an epidemic model with non-monotone “force of infection”.     

New periodic and soliton solutions for the Generalized BBM and Burgers-BBM equations

In this paper, we consider the Benjamin Bona Mahony equation (BBM), and we obtain new exact solutions for it by using a generalization of the well-known tanh-coth method. New periodic and soliton solutions for the Generalized BBM and Burgers-BBM equations are formally derived.     

Traveling waves for delayed non-local diffusion equations with crossing-monostability

This paper is concerned with the traveling waves for a class of delayed non-local diffusion equations with crossing-monostability. Based on constructing two associated auxiliary delayed non-local diffusion equations with quasi-monotonicity and a profile set in a suitable Banach space using the traveling wave fronts of the auxiliary equations, the existence of traveling waves is proved by Schauder’s fixed point theorem. The result implies that the traveling waves of the delayed non-local diffusion equations with crossing-monostability are persistent for all values of the delay τ?0.     

Traveling wave solutions in delayed reaction-diffusion systems with mixed monotonicity

This paper deals with the existence of traveling wave solutions in delayed reaction-diffusion systems with mixed monotonicity. Based on two different mixed-quasi monotonicity reaction terms, we propose new conditions on the reaction terms and 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 diffusive Lotka-Volterra systems.     

时滞偏微分方程行波解的持久性

本文利用扰动法、Fredholm理论及经典的不动点定理,研究了时滞偏微分方程行波解的存在性.我们的结果表明,对于没有时滞时任意有意义的波速,在小时滞扰动下行波解具有持久性.     

On soliton solutions for a generalized Hirota–Satsuma coupled KdV equation

The extended homogeneous balance method is used to construct exact traveling wave solutions of a generalized Hirota–Satsuma coupled KdV equation, in which the homogeneous balance method is applied to solve the Riccati equation and the reduced nonlinear ordinary differential equation, respectively. Many exact traveling wave solutions of a generalized Hirota–Satsuma coupled KdV equation are successfully obtained, which contain soliton-like and periodic-like solutions This method is straightforward and concise, and it can also be applied to other nonlinear evolution equations.     

The Delivery Man Problem with time windows

In this paper, a variant of the Traveling Salesman Problem with Time Windows is considered, which consists in minimizing the sum of travel durations between a depot and several customer locations. Two mixed integer linear programming formulations are presented for this problem: a classical arc flow model and a sequential assignment model. Several polyhedral results are provided for the second formulation, in the special case arising when there is a closed time window only at the depot, while open time windows are considered at all other locations. Exact and heuristic algorithms are also proposed for the problem. Computational results show that medium size instances can be solved exactly with both models, while the heuristic provides good quality solutions for medium to large size instances.