Kink-soliton, singular-kink-soliton and singular-periodic solutions for a new two-mode version of the Burger–Huxley model: applications in nerve fibers and liquid crystals

New two-mode version of the generalized Burger–Huxley equation is derived using Korsunsky’s operators. The new model arises in the applications of nerve fibers and liquid crystals, and it describes the interaction of two symmetric waves moving simultaneously in the same direction. Solitary wave solutions of types kink-soliton, singular-kink-soliton and singular-periodic are obtained to this model by means of the simplified bilinear method, polynomial-function method and the Kudryashov-expansion method. A comprehensive graphical analysis is conducted to show some physical properties of this new type of nonlinear equations. Finally, all obtained solutions are verified by direct substitutions in the new model.