On a generalized nonlinear Korteweg – De Vries equation

Abstract

Background. The study provides a detailed research of the generalized Korteweg – De Vries equation, obtained using the method of least action, which indicates the relevance of the study, the purpose of which is to analyze the resulting equation. Materials and methods. The main research method is the variational approach. Results. The main result of the work is the analytical derivation of the generalized Korteweg – De Vries equation. Conclusions. A general view of the invariant functional of classical action (1) in the language of pseudovector A x( ) which makes it possible to solve a certain class of problems from the field of nonlinear dynamics. A detailed analysis of the functional (1) in the case of one-dimensional motion, when the vector function A x( ) leads to the functional (4), is given. Using the method of least action, a generalized equation of the dynamics of a one-dimensional system (6) is obtained. In some special cases, a detailed analysis of the solutions of equation (6) is given, which makes it possible in particle cases automatically describe two physically important results: a solitary wave and a shock wave front.

Keywords

• action functional
• variation
• korteweg – de vries equation
• nonlinear differential equation