Equivalence Groupoid and Exact Solutions of a Class of Generalized Modified Korteweg–de Vries Equations.

Abstract

The transformation properties of a class of generalized modified Korteweg–de Vries equations with coefficients dependent on the time variable are investigated, demonstrating the effectiveness of the equivalence method in constructing exact solutions for such equations. Specifically, the equivalence groupoid for the class is identified, and it is proven that the class is normalized. A criterion for the reducibility of equations with variable coefficients from the class to the classical modified Korteweg–de Vries equation is established. The study reveals that the equivalence method for finding exact solutions is more effective for such equations than other available methods. Consequently, a formula for generating exact solutions of the generalized modified Korteweg–de Vries equations with variable coefficients is derived, and examples of constructing exact solutions are presented.