Mathematical tricks for pseudopotentials in the theories of nonlinear waves in plasmas

Abstract

In the analysis of nonlinear waves in plasma, especially for the search for periodic waves, shock waves, and solitons, mechanical analogy methods are widely applicable. The most famous of them is the Sagdeev pseudopotential method. However, sometimes mathematical difficulties arise when deriving formulas for pseudopotentials. The author proposes three mathematical tricks to get around these difficulties and obtain exact formulas for pseudopotentials in cases where the direct, Sagdeev method is considered inapplicable: a trick based on the Lambert W-function, a trick based on the inverse function integration, and a trick based on reducing the theory equations to the Bernoulli differential equation (the Bernoulli pseudopotential method). This article, which is methodological by nature, provides detailed examples of the application of each of these tricks when deriving formulas for pseudopotentials.