A Note on the Propagation Speed of Travelling Waves for a Lotka–Volterra Competition Model with Diffusion

We derive the John–Sclavounos equations, describing the motion of a fluid particle on the sea surface, from first principles using Lagrangian and Hamiltonian formalisms applied to the motion of a frictionless particle constrained on an unsteady surface. This framework leads to a number of new insights into the particle kinematics. The main result is that vorticity generated on a stress-free surface vanishes at a wave crest when the horizontal particle velocity equals the crest propagation speed, which is the kinematic criterion for wave breaking. If this holds for the largest crest, then the symplectic two-form associated with the Hamiltonian dynamics reduces instantaneously to that associated with the motion of a particle in free flight, as if the surface did not exist. Further, exploiting the conservation of the Hamiltonian function for steady surfaces and travelling waves, we show that particle velocities remain bounded at all times, ruling out the possibility of the finite-time blowup of solutions.

Download Full-text

Parameter Dependence of Propagation Speed of Travelling Waves for Competition-Diffusion Equations

SIAM Journal on Mathematical Analysis ◽

10.1137/s0036141093244556 ◽

1995 ◽

Vol 26 (2) ◽

pp. 340-363 ◽

Cited By ~ 82

Author(s):

Yukio Kan-on

Keyword(s):

Travelling Waves ◽

Propagation Speed ◽

Diffusion Equations ◽

Parameter Dependence ◽

Competition Diffusion

Download Full-text

Note on propagation speed of travelling waves for a weakly coupled parabolic system

Nonlinear Analysis ◽

10.1016/s0362-546x(99)00261-8 ◽

2001 ◽

Vol 44 (2) ◽

pp. 239-246 ◽

Cited By ~ 5

Author(s):

Yukio Kan-On

Keyword(s):

Parabolic System ◽

Travelling Waves ◽

Propagation Speed ◽

Weakly Coupled

Download Full-text

The development of travelling waves in quadratic and cubic autocatalysis with unequal diffusion rates. I. Permanent form travelling waves

Philosophical Transactions of the Royal Society of London Series A Physical and Engineering Sciences ◽

10.1098/rsta.1991.0001 ◽

1991 ◽

Vol 334 (1633) ◽

pp. 1-24 ◽

Cited By ~ 86

Keyword(s):

Travelling Waves ◽

Chemical Species ◽

Propagation Speed ◽

Reaction Diffusion ◽

Spatial Dimension ◽

Travelling Wave ◽

Diffusion Waves ◽

Long Time ◽

Diffusion Rates ◽

Cubic Autocatalysis

We study the isothermal autocatalytic system , A + n B → ( n + 1)B , where n = 1 or 2 for quadratic or cubic autocatalysis respectively. In addition, we allow the chemical species, A and B, to have unequal diffusion rates. The propagating reaction-diffusion waves that may develop from a local initial input of the autocatalyst, B, are considered in one spatial dimension. We find that travelling wave solutions exist for all propagation speeds v ≥ v * n ,where v * n is a function of the ratio of the diffusion rates of the species A and B and represents the minimum propagation speed. It is also shown that the concentration of the autocatalyst, B, decays exponentially ahead of the wavefront for quadratic autocatalysis. However, for cubic autocatalysis, although the concentration of the autocatalyst decays exponentially ahead of the wavefront for travelling waves which propagate at speed v = v * 2 , this rate of decay is only algebraic for faster travelling waves with v > v * 2 . This difference in decay rate has implications for the selection of the long time wave speed when such travelling waves are generated from an initial-value problem.

Download Full-text