Spreading speed and traveling waves for a nonmonotone reaction–diffusion model with distributed delay and nonlocal effect

In this paper, spreading speed and traveling waves for reaction–diffusion model with distributed delay and nonlocal effect without monotonicity are investigated. It is shown that there exists the spreading speed c^∗ which coincides with the minimal wave speed, and its limiting integral equation has an unique traveling wave with speed c > c^∗, and no traveling wave with c < c^∗. Moreover, the dependence of the spreading speed on the delay and the nonlocal effect is considered.     

The evolution of traveling waves in a simple isothermal chemical system modeling quadratic autocatalysis with strong decay

In this paper, we study a reaction–diffusion system for an isothermal chemical reaction scheme governed by a quadratic autocatalytic step A+B→2B$A+B\to 2B$ and a decay step B→C$B\to C$, where A, B, and C are the reactant, the autocatalyst, and the inner product, respectively. Previous numerical studies and experimental evidences demonstrate that if the autocatalyst is introduced locally into this autocatalytic reaction system where the reactant A initially distributes uniformly in the whole space, then a pair of waves will be generated and will propagate outwards from the initial reaction zone. One crucial feature of this phenomenon is that for the strong decay case, the formation of waves is independent of the amount of the autocatalyst B introduced into the system. It is this phenomenon of KPP-type which we would like to address in this paper. To study the propagation of reactant and autocatalyst analytically, we first use the tail behavior of waves to construct a pair of generalized super-/sub-solutions for the approximate system of the autocatalytic reaction system. Note that the autocatalytic reaction system does not enjoy comparison principle. Together with a family of truncated problems, we can establish the existence of a family of traveling waves with the minimal speed. Second, we use this pair of generalized super-/sub-solutions to show that the propagation of waves is fully determined by the rate of decay of the initial data at infinity in the sense of Aronson–Weinberger formulation, which in turn confirms the aforementioned numerical and experimental results.     

A continuous variable representation of the traveling salesman problem

The classic traveling salesman problem is characterized in terms of continuous flows on a specially constructed non-conservative network, in 2n – 1 linear constraints and a cardinality constraint. It is shown that every solution to the network problem is a hamiltonian circuit.     

Radiation Losses of Bent Planar Waveguides

Bent planar waveguides are conformally mapped into straight figures with equivalent index profile. Airy functions of traveling-wave form are introduced to express the radiation mode in the outer cladding region. Calculation results of radiation loss are confirmed to give excellent accuracy.     

Nonlinear waves in electromigration dispersion in a capillary

We construct exact solutions to an unusual nonlinear advection–diffusion equation arising in the study of Taylor–Aris (also known as shear) dispersion due to electroosmotic flow during electromigration in a capillary. An exact reduction to a Darboux equation is found under a traveling-wave ansatz. The equilibria of this ordinary differential equation are analyzed, showing that their stability is determined solely by the (dimensionless) wave speed without regard to any (dimensionless) physical parameters. Integral curves, connecting the appropriate equilibria of the Darboux equation that governs traveling waves, are constructed, which in turn are shown to be asymmetric kink solutions (i.e., non-Taylor shocks). Furthermore, it is shown that the governing Darboux equation exhibits bistability, which leads to two coexisting non-negative kink solutions for (dimensionless) wave speeds greater than unity. Finally, we give some remarks on other types of traveling-wave solutions and a discussion of some approximations of the governing partial differential equation of electromigration dispersion.