A HAM-based analytic approach for physical models with an infinite number of singularities

Abstract

Based on the Homotopy Analysis Method (HAM), an analytic approach is proposed to solve physical models with an infinite number of “singularities”. The nonlinear interaction of double cnoidal waves governed by the Korteweg-de Vries (KdV) equation is used to illustrate its validity. The HAM is an analytic technique for highly nonlinear problems, which is based on the homotopy in topology and thus has nothing to do with small physical parameters. Besides, the HAM provides us great freedom to choose proper equation-type and solution-expression for high-order approximation equations. Especially, unlike other methods, the HAM can guarantee the convergence of solution series. Using the HAM, an infinite number of zero denominators of the considered problem are avoided once for all by properly choosing an auxiliary linear operator, as illustrated in this paper. This HAM-based approach has general meanings and can be used to solve many physical problems with lots of “singularities”. It also suggests that the so-called “singularity” might not exist physically, but only due to the imperfection of used mathematical methods, because the nature should not contain any singularities at all.