Towards an approximation of solitary-wave solutions of non-integrable evolutionary pdes via symmetry and qualitative analysis

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 dʹAlembert equation and the hyperbolic generalization of the Burgers equation.