On solutions of one of the second-order nonlinear differential equation: An in-depth look and critical review

Abstract

A critical review of recent articles by two scientific groups, which have considered a well-known nonlinear differential equation of the second order, is presented. One of these groups is led by G. Akram et. al. from Pakistan (Department of mathematics, University of the Punjab, Lahore). Another group is led by K.-J. Wang from China (School of Physics and Electronic Information Engineering, Henan Polytechnic University, Jiaozuo). In a number of papers published by these authors there have been presented a lot of solutions of the well-known differential equation. In fact, this differential equation was studied more than 150 years ago in the works of outstanding mathematicians Niels Henrik Abel (1827), Karl Gustav Jacob Jacobi (1829) and Karl Weierstrass (1855, 1862). However, the scientific groups of Akram and Wang, apparently not being familiar with the works of prominent mathematicians and not realizing that this equation has a unique solution on the complex plane, have been trying to rewrite the solution of this equation using symbolic mathematics programs misleading by that the scientific community. Although there are several erroneous works by Akram and Wang, only a few articles are analyzed here. The errors of a few works by these authors are discussed. The correct solutions of this popular equation, which is often encountered in nonlinear optics, are presented.