Travelling Waves in Nonlinear Diffusion-Convection-Reaction Processes
by Róbert Kersner
Special solutions play an important role in the study of nonlinear
partial differential equations (PDE's) arising in mathematical biology.
Confronted with a mathematical model in the form of an initial or a boundary
value problem for a partial differential equation or a system of such equations,
the foremost desire of a practitioner is to solve the problem explicitly.
If little theory is available and no explicit solutions are obtainable,
generally the ensuing attack is to identify circumstances under which the
complexity of the problem may be reduced. The study of these problems belongs
to the main research directions of the Applied Mathematics Laboratory at
SZTAKI.
In cooperation with the University of Twente, a technique has been developed
for determining whether or not a nonlinear, possibly degenerate parabolic
partial differential equation admits a travelling-wave (TW) solution, ie
solution of the form f(x-ct) which does not change the shape f and where
the speed c is constant. This technique is suitable for investigating properties
of such travelling waves as well.
In ecological context, the different terms in such kind of partial differential
equations represent the birth - death process, diffusion and convection.
Typical examples for them are the Fisher or logistic equation, which is
the archetypical deterministic model for the spread of an advantageous
gene in a population of diploid individuals, the Nagumo equation, which
models the transmission of electrical pulses in a nerve axon, and the Richards
(or nonlinear Fokker-Planck) equation. Analogous systems of PDEs were proposed
as models for the chemical basis of morphogenesis by Turing in 1952.
The principal contention of our method is that the study of the travelling
wave solutions of partial differential equations above is equivalent to
the study of the singular Volterra-type nonlinear integral equations in
the following sense: the partial differential equation has a monotonic
travelling wave solution connecting two states with a given speed if and
only if the integral equation has a non-negative solution which is zero
at these states.
For the Fisher-type equations a typical result is that the travelling
wave solutions exist only for speeds which are bigger than the minimal
speed and have the form given in Figure 1. The minimal speed is an unknown
of the problem, having great practical importance. In many cases, after
some time the general process moves with this velocity.
It may happen that a travelling wave solution has a sharp front (see
Figure 2). This is the case, for example, for the generalised Fischer-type
model, when the density dependence of the diffusion coefficient is power-like.
The travelling wave solutions with not minimal speed have no sharp front
(see Figure 1), but the unique wave with minimal speed has (see Figure
2). Note that this travelling wave has attractor property: after some time
all the solutions of the original partial differential equation with initial
data from a given class will take the form and the speed of this wave.
Using the above mentioned integral equation method one can obtain these
kind of results almost algorithmically, together with good estimates for
the minimal speed.
Our developed technique provides an alternative to phase-plane analysis
and has several advantages with respect to this classical approach. Turning
to the results of the theory of integral equations the user can easily
identify circumstances which guarantee the existence of the required waves
and can decide if the wave has a sharp front or not. Models with the same
characteristics arise in several scientific fields, for example heat transfer,
combustion, reaction chemistry, plasma physics, etc.
Please contact:
Róbert Kersner - SZTAKI
Tel: +36 1 4665 783
E-mail: kersner@sztaki.hu
