Simple equations method: Methodology, inspiration by the research of Kudryashov, and several remarks on the application of balance equations

We discuss an aspect of the application of the Simple Equations Method (SEsM) for obtaining exact solutions of nonlinear differential equations. The aspect is related to the number of balance equations needed to obtain an exact solution of nonlinear differential equations solved. The work and results of Professor Kudryashov stimulated our research on SEsM at an important period in the development of this methodology. Because of this, we start with a short description of SEsM and then briefly review our research on the exact solution of nonlinear differential equations, as well as some of the results of Prof. Kudryashov in this area in the last 30 years. We apply the specific case $$\mathrm{SEsM}(1,1)$$ of the SEsM to the following class of nonlinear differential equations: $$\sum_{f=0}^{f_{\max}}\sum_{\omega=1}^{n}\sum_{\omega_1=0}^\omega A_{f,\omega,\omega_1} \biggl(F,\biggl\{\frac{\partial^{\zeta}F}{\partial x^{\zeta_1}\,\partial t^{\zeta-\zeta_1}}\biggr\}\biggr) \biggl[\frac{\partial^\omega F}{\partial x^{\omega_1}\,\partial t^{\omega-\omega_1}}\biggr]^f=B(F),$$ where $$A_{f,\omega,\omega_1}\bigl(F,\bigl\{\frac{\partial^{\zeta}F}{\partial x^{\zeta_1}\partial t^{\zeta-\zeta_1}}\bigr\}\bigr)$$ and $$B(F)$$ are polynomials in the unknown function $$F$$ and its derivatives. As a simple equation, we use an ordinary differential equation $$\bigl(\frac{d\Phi}{d\xi}\bigr)^\epsilon=\sum_{\pi=0}^{\sigma}\gamma_{\pi}[\Phi (\xi)]^\pi$$ , which contains as a specific case, the elliptic equation $$\bigl(\frac{d\Phi}{d\xi}\bigr)^2=a\Phi^4+b\Phi^2+c$$ . We show that this can lead to the necessity of using more than one balance equation. The methodological results are illustrated by selected simple examples.