Variational Principles for Some Nonlinear Wave Equations

Generally speaking, there exist two basic ways to describe a physical problem [1]: (1) by differential equations (DE) with boundary or initial conditions; (2) by variational principles (VP). The VP model has many advantages over its DE partner: simple and compact in form while comprehensive in content, encompassing implicitly almost all information characterizing the problem under consideration [1 – 5]. Variational methods have been, and continue to be, popular tools for nonlinear analysis. When contrasted with other approximate analytical methods, variational methods combine the following two advantages: (1) they provide physical insight into the nature of the solution of the problem; (2) the obtained solutions are the best among all the possible trial functions. The variational-based analytical methods, e. g., the variational iteration method [6 – 11] and He’s variational method [1], have become hot topics in recent publications. Although the variational principles of fluid dynamics [1 – 5] have been studied for a long time, yet the general variational principles of various nonlinear wave equations have not been dealt with systematically. In this paper we illustrate how to establish a variational formulation for a nonlinear problem using the semi-inverse method proposed by Ji-Huan He [2].