DOI link for Introduction
Nonlinear partial differential equations (PDE) are widely used in mathematical modeling real world processes since the beginning of the 20th century only. Probably, one of the attempts in applying and solving a nonlinear PDE of the parabolic type was made by J. Boussinesq who studied the porous diffusion equation describing the water filtration in soil. The most widely used type of equations for modeling such processes are the nonlinear reaction-diffusion-convection (RDC) equations. As a result, in the 1970sseveral monographs were published, which are devoted to study and application of the nonlinear RDC equations in physics, biology and chemistry. The well-known principle of linear superposition cannot be applied to generate new exact solutions to nonlinear PDEs. The most powerful methods for construction of exact solutions for a wide range of classes of nonlinear PDEs are symmetry-based methods. All these methods have the common idea stating that exact solutions can be found for a given PDE provided its symmetry is known.