An overview of recently obtained authors’ results on traveling wave solutions of some
classes of PDEs is presented. The main aim is to describe all possible travelling wave
solutions of the equations. The analysis was conducted using the methods of qualitative
and bifurcation analysis in order to study the phase-parameter space of the corresponding
wave systems of ODEs. In the first part we analyze the wave dynamic modes of populations
described by the “growth - taxis - diffusion" polynomial models. It is shown that
“suitable" nonlinear taxis can affect the wave front sets and generate non-monotone waves,
such as trains and pulses, which represent the exact solutions of the model system.
Parametric critical points whose neighborhood displays the full spectrum of possible model
wave regimes are identified; the wave mode systematization is given in the form of
bifurcation diagrams. In the second part we study a modified version of the
FitzHugh-Nagumo equations, which model the spatial propagation of neuron firing. We assume
that this propagation is (at least, partially) caused by the cross-diffusion connection
between the potential and recovery variables. We show that the cross-diffusion version of
the model, besides giving rise to the typical fast travelling wave solution exhibited in
the original “diffusion" FitzHugh-Nagumo equations, additionally gives rise to a slow
traveling wave solution. We analyze all possible traveling wave solutions of the model and
show that there exists a threshold of the cross-diffusion coefficient (for a given speed
of propagation), which bounds the area where “normal" impulse propagation is possible. In
the third part we describe all possible wave solutions for a class of PDEs with
cross-diffusion, which fall in a general class of the classical Keller-Segel models
describing chemotaxis. Conditions for existence of front-impulse, impulse-front, and
front-front traveling wave solutions are formulated. In particular, we show that a
non-isolated singular point in the ODE wave system implies existence of free-boundary
fronts.