Towards approximation of solitary-wave solutions of non-integrable evolutionary PDEs, via symmetry and qualitative analysis

Abstract:
It is well known that various wave patterns, observed in open dissipative systems, are described by the non-linear PDE, being not completely integrable. For this reason their analytical description, generally speaking, is rather impossible. Yet the information about the existence of the wave patterns among the solutions with the given symmetry can always be obtained by means of qualitative theory methods. Such synthetic approach proves to deliver sufficient information for finding out approximated solitary wave regimes. We test the effectiveness of this algorithm on the  non-linear d'Alembert equation and the hyperbolic generalization of the Burgers equation.