Variational Approach To Ion Acoustic Waves Equation

With the development of the computer science, there exits no difficulty in solving linear equations now, however, we have no universal approach to nonlinear equations especially those with strong nonlinearity. The nonlinear science has been caught much attention since 1960s, and the nonlinear equations are becoming richer and richer. The nonlinear equations have some special characters, and various different methods with merits on the one hand and disadvantages on the other hand are introduced to solve various nonlinear equations.In this thesis, we apply the variational method to the ion acoustic wave equations, and it is shown to be a great success. In addition to the tremendous practical importance of variational principles in establishing approximate methods, there are some reasons of a more fundamental nature which motivate variational formulation in science.As it is well known that there exists no variational formulationfor differential equations with first-order derivatives which occurs in most nonlinear wave equations. In order to overcome the difficulty, a transformation is introduced. The transformed equations admit a variational formulation.Chapter 1 introduces some basic characters of the variational principle. It is always difficulty to establish a variational principle for nonlinear equations, some examples are given in Chapter 2 to illustrate the way how to construct variational formulations direct from the field equations. It is shown that the semi-inverse method is very effective and convenient to construct variational formulae of some nonlinear equations, e.g., Yang-Mill equation, the nonlinear electrochemical equation and others. In chapter 3, various approximate analytical methods including classical perturbation method and reductive perturbation method are introduced. Chapter 4 discusses the establishment of a variational formulation for the well-known KdV equation. The Ritz method is applied to solve its approximate solitary solution. Comparing of the approximate solution with the exact one reveals the effectiveness of the variational method. Chapter 5 discusses the nonlinear ion acoustic wave equations in details. In order to establish a variational formulation, a special function is introduced. A generalizedvariational principle is established. Based on the established variational theory, a new approximate solitary solution is obtained by Ritz method, revealing the relationship among the velocity, height and the solitary width of the wave.