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We study a system of differential equations simulating transport phenomena in active structured media. The model is a generalization of McKean's modification of the celebrated FitzHugh–Nagumo system, describing the nerve impulse propagation in axon. It takes into account the effects of memory, connected with the presence of internal structure. We construct explicitly the localized traveling wave solutions and analyze their stability.

Potential symmetry of a class of nonlinear transport equations taking into account the effects of memory is studied. For a specific transport coefficient the symmetry is shown to be infinite. This fact is used for constructing nonlocal transformation linearizing the transport equation. New formulae of nonlocal nonlinear superposition and generation of solutions are proposed. Additional Lie symmetries of the corresponding linear equations are used to construct nonlocal symmetries of the source equation. The formulae derived are used for the construction of exact solutions.

Spectral stability of the kink-like and soliton-like traveling wave solutions to the generalized convection-reaction-diffusion equation is studied by means of the qualitative methods and numerical simulations.

It is well known that various wave patterns observed in open dissipative systems are described by nonlinear PDEs, being not, as a rule, completely integrable. Yet the information about the existence of solutions describing the wave patterns (periodic, kink-like, soliton-like regimes and so on) within the dissipative model can be obtained by means of qualitative theory methods. In this work we show how it is possible, using the self-similar reduction and the qualitative analysis, to find approximated solutions to evolutionary PDEs, describing the solitary wave regimes. We apply this approach to the nonlinear d'Alembert equation and the hyperbolic generalization of the Burgers equation.