On symmetry reductions, solutions and conservation laws for a one-dimensional third-order Korteweg–de Vries equation with power law nonlinearity

Abstract

This work investigates a general form of one-dimensional Korteweg–de Vries (KdV) equation which arises in mathematical sciences. The model is examined for the first time with the use of techniques such as the symmetry method, direct integration procedure and explicit power series method. These methods provided us with new closed-form solutions of the one-dimensional third-order Korteweg–de Vries equation (1.3) in different forms. Each method employed here has its own merits. In addition, we include 2D and 3D graphical representations of selected exact solutions for specific parameter values to improve the reader’s understanding of these findings. Besides, for the first time conservation laws are presented in this work. We employ two different approaches, namely Noether’s theorem and multiplier method. The conserved vectors obtain contains conservation of momentum and energy.