•Concept of polynomials in logistic function is introduced.•Nonlinear differential equations with exact solutions in the form of polynomials in logistic function are found.•It is shown there are solitary waves with many maximum and minimum.•Algorithm for finding of exact solutions by means of polynomial in logistic function is given. Properties of polynomials in logistic function are studied. It is demonstrated that these polynomials can be used for construction of exact solutions to nonlinear differential equations. Nonlinear differential equations with exact solutions in the form of polynomials in logistic function are found. It is shown there are solitary waves of nonlinear differential equations described by polynomial in logistic function with many maximum and minimum.