# Simple Equations Method and Non-Linear Differential Equations with Non-Polynomial Non-Linearity

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^{2}

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## Abstract

**:**

## 1. Introduction

## 2. The Simple Equations Method (SEsM)

**Step****(1.)****Transformation of the non-linearity of the solved equation**We apply the following transformations:$$u(x,\dots ,t)=T[{F}_{1}(x,\cdots ,t),\cdots ,{F}_{N}(x,\cdots ,t)].$$In numerous cases, one may skip this step (then we have $u(x,\dots ,t)=F(x,\dots ,t)$). In many other cases, the transformation is needed for obtaining a solution of the studied non-linear PDE. The application of (2) to (1) leads to non-linear differential equations for the functions ${F}_{i}$. We do not know the general form for the transformation T. The reason is that the non-linearity in the solved equations can be of different kinds.We note that Step (1.) of SEsM will be at the focus of our study in this article. We are going to study non-linear equations for which non-linearity can be reduced to polynomial non-linearity by means of appropriate transformations.**Step****(2.)****The solution is searched as composite function of solutions of more simple equations**In this step, the functions ${F}_{i}(x,\dots ,t)$ are chosen as composite functions of functions ${f}_{1},\dots $, which are solutions of more simple differential equations. In general, we do not fix the relationship for the composite function. Then, we use the general Faa di Bruno relationship for the derivatives of a composite function [113]. In MMSE, we have used a fixed relationship for the composite function. For an example, for the case of 1 solved equation and one function F:$$\begin{array}{c}\hfill F=\alpha +\sum _{{i}_{1}=1}^{N}{\beta}_{{i}_{1}}{f}_{{i}_{1}}+\sum _{{i}_{1}=1}^{N}\sum _{{i}_{2}=1}^{N}{\gamma}_{{i}_{1},{i}_{2}}{f}_{{i}_{1}}{f}_{{i}_{2}}+\sum _{{i}_{1}=1}^{N}\cdots \sum _{{i}_{N}=1}^{N}{\sigma}_{{i}_{1},\cdots ,{i}_{N}}{f}_{{i}_{1}}\cdots {f}_{{i}_{N}}.\end{array}$$**Step****(3.)****Selection of the simple equations**We select the simple equations for the functions ${f}_{1},\dots $. In addition, we have to fix the relationship between the composite functions ${F}_{i}(x,\dots ,t)$ and the functions ${f}_{1},\dots $. We note that the fixation of the simple equations and the fixation of the relationships for the composite functions are connected. The reason for this is as follows. The fixations transform the left-hand sides of the solved equations. The result of this transformation can be functions which are sums of terms. Each term contains some function multiplied by a coefficient. The coefficient is a relationship connecting some of the parameters of the solved equations and some of the parameters of the solutions of the used simple equations. Each coefficient must have at least two terms (Otherwise, the trivial solution will be produced). In order to ensure this, a balance procedure must be applied. This balance procedure leads to one or more additional relationships among the parameters of the solved equation and parameters of the solutions of the used simple equations. The additional relationships are called balance equations. The balance equations are the connection between the choice of the simple equation and the fixation of the form of the composite function.**Step****(4.)****Solution of the obtained system of non-linear algebraic equations**We may obtain a nontrivial solution of (1) if all coefficients mentioned in Step (3.) are set to 0. This condition leads to a system of non-linear algebraic equations. The equations connect the coefficients of the solved non-linear differential equation and for the coefficients of the solutions of the simple equations. Any nontrivial solution of this algebraic system leads to a solution of the studied non-linear partial differential equation.There are two possibilities for the solution of the system of non-linear algebraic equations:- The number which is the sum of the number of parameters of the solution and the number of parameters of the equation can be larger than the number of algebraic equations or equal to the number of algebraic equations. Then, the system usually (but not in all of the cases) has a nontrivial solution(s). Independent parameters may be presented in this situation. The other parameters of the solution are functions of these independent parameters.
- The number which is the sum of the number of parameters of the solution and the number of parameters of the equation is smaller than the number of algebraic equations. Then, the system of algebraic equations usually does not have a nontrivial solution. However, there can be important exceptions to this. An exception occurs when the number of equations of the algebraic system can be reduced and this number becomes less or equal to the number of available parameters. Then, this case is reduced to the previous one and a nontrivial solution is possible.

## 3. Several Transformations Which Are of Interest for the SEsM

#### 3.1. General Considerations

**Proposition**

**1.**

- Terms containing only derivatives of u;
- Terms containing one or several non-polynomial non-linearities of the function u and these non-polynomial non-linearity are of the same kind.

- Property 1: The transformation T transforms any of the non-polynomial non-linearity to a function which contains only polynomials of F.
- Property 2: The transformation T transforms the derivatives of u to terms containing only polynomials of derivatives of F or polynomials of derivatives of F multiplied or divided by polynomials of F.

**Proof.**

#### 3.2. Several Kinds of Non-Linearity Possessing the Properties 1 and 2 from the Proposition above

**Case****1:**- $N\left(u\right)=exp\left(u\right)$; $N\left(u\right)={[exp\left(u\right)]}^{m}$In this case, the transformation is $u=ln\left(F\right)$. Let us consider first the case $N\left(u\right)=exp\left(u\right)$. The transformation has Property 1 as follows:$$N\left(u\right)=exp\left(u\right)=exp[ln(F\left)\right]=F.$$$${u}_{x}={[ln\left(F\right)]}_{x}=\frac{{F}_{x}}{F}.$$$${u}_{xx}=\frac{{F}_{xx}}{F}-\frac{{F}_{x}^{2}}{{F}^{2}};\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{u}_{xt}=\frac{{F}_{xt}}{F}-\frac{{F}_{x}{F}_{t}}{{F}^{2}}.$$
**Case****2:**- $N\left(u\right)=sin\left(u\right)$; $N\left(u\right)={[sin\left(u\right)]}^{m}$.In this case, a possible transformation is $u=4{tan}^{-1}\left(F\right)$. Let us consider first the case $N\left(u\right)=sin\left(u\right)$. The transformation has Property 1 as follows:$$N\left(u\right)=sin\left[4{tan}^{-1}\left(F\right)\right]=4\frac{F(1-{F}^{2})}{{(1+{F}^{2})}^{2}}$$$${u}_{x}=\frac{4}{1+{F}^{2}}{F}_{x}$$$$N\left(u\right)={\{sin\left[4{tan}^{-1}\left(F\right)\right]\}}^{m}={4}^{m}\frac{{\left[F(1-{F}^{2})\right]}^{m}}{{\left[{(1+{F}^{2})}^{2}\right]}^{m}}.$$
**Case****3:**- $N\left(u\right)=cos\left(u\right)$; $N\left(u\right)={[cos\left(u\right)]}^{m}$.We consider first the case $N\left(u\right)=cos\left(u\right)$. The transformation is $u=4{tan}^{-1}\left(F\right)$. The transformation has Property 1 as follows:$$N\left(u\right)=\frac{{(1-{F}^{2})}^{2}-4{F}^{2}}{{(1+{F}^{2})}^{2}}$$$${u}_{x}=\frac{4}{1+{F}^{2}}{F}_{x}.$$$$N\left(u\right)=\frac{{[{(1-{F}^{2})}^{2}-4{F}^{2}]}^{m}}{{\left[{(1+{F}^{2})}^{2}\right]}^{m}}$$
**Case****4:**- $N\left(u\right)=tan\left(u\right)$; $N\left(u\right)={[tan\left(u\right)]}^{m}$.We first consider the case $N\left(u\right)=tan\left(u\right)$. In this case, a possible transformation is $u={tan}^{-1}\left(F\right)$. The non-linearity is transformed to a polynomial of F: $N\left(u\right)=tan\left[{tan}^{-1}\left(F\right)\right]=F$. The derivative of u is also reduced to a relationship containing the polynomial of F and a derivative of F. For an example:$${u}_{x}=\frac{1}{1+{F}^{2}}{F}_{x}.$$
**Case****5:**- $N\left(u\right)=cot\left(u\right)$; $N\left(u\right)={[cot\left(u\right)]}^{m}$.The transformation in this case is $u={cot}^{-1}\left(F\right)$. The non-linearity is transformed to a polynomial of F: $N\left(u\right)=cot\left[{cot}^{-1}\left(F\right)\right]=F$. The derivative of u is reduced to a relationship containing a polynomial of F and a derivative of F. For an example:$${u}_{x}=-\frac{1}{1+{F}^{2}}{F}_{x}$$
**Case****6:**- $N\left(u\right)=sinh\left(u\right)$; $N\left(u\right)={[sinh\left(u\right)]}^{m}$.In this case, the transformation is $u=4{tanh}^{-1}\left(F\right)$. The derivatives of u contain derivatives of F and polynomials of F. For an example:$${u}_{x}=\frac{4}{1-{F}^{2}}{F}_{x}$$$$N\left(u\right)=sinh\left[4{tanh}^{-1}\left(F\right)\right]=4\frac{F(1+{F}^{2})}{{(1-{F}^{2})}^{2}}$$$$N\left(u\right)={4}^{m}\frac{{\left[F(1+{F}^{2})\right]}^{m}}{{\left[{(1-{F}^{2})}^{2}\right]}^{m}}$$
**Case****7:**- $N\left(u\right)=cosh\left(u\right)$; $N\left(u\right)={[cosh\left(u\right)]}^{m}$.In this case, the transformation is $u=4{tanh}^{-1}\left(F\right)$. For the case $N\left(u\right)=cosh\left(u\right)$:$$N\left(u\right)=\frac{{(1+{F}^{2})}^{2}+4{F}^{2}}{{(1-{F}^{2})}^{2}}$$$${u}_{x}=\frac{4}{1-{F}^{2}}{F}_{x}$$For the case $N\left(u\right)={[cosh\left(u\right)]}^{m}$, the transformation has the Property 1 as follows:$$N\left(u\right)=\frac{{[{(1+{F}^{2})}^{2}+4{F}^{2}]}^{m}}{{\left[{(1-{F}^{2})}^{2}\right]}^{m}}$$
**Case****8:**- $N\left(u\right)=tanh\left(u\right)$; $N\left(u\right)={[tanh\left(u\right)]}^{m}$.In this case, the transformation is $u\left(F\right)={tanh}^{-1}\left(F\right)$. $N\left(u\right)$ is reduced to $N\left(u\right)=F$, which is a polynomial of F. The derivatives of u contains polynomials of F and derivatives of F. For an example:$${u}_{x}=\frac{1}{1-{F}^{2}}$$
**Case****9:**- $N\left(u\right)=coth\left(u\right)$; $N\left(u\right)={[coth\left(u\right)]}^{m}$.In this case, the transformation is $u\left(F\right)={coth}^{-1}\left(F\right)$. $N\left(u\right)$ is reduced to $N\left(u\right)=F$, which is a polynomial of F. The derivatives of u contains polynomials of F and derivatives of F. For an example:$${u}_{x}=-\frac{1}{{F}^{2}-1}$$
**Case****10:**- $N\left(u\right)=sin\left(mu\right)$; $N\left(u\right)=cos\left(mu\right)$.In this case, we can use the following relationships:$$sin\left(mu\right)=\sum _{k=0}^{m}\left(\genfrac{}{}{0pt}{}{m}{k}\right){cos}^{k}\left(u\right){sin}^{m-k}\left(u\right)sin\left[\frac{\pi}{2}(m-k)\right]$$$$cos\left(mu\right)=\sum _{k=0}^{m}\left(\genfrac{}{}{0pt}{}{m}{k}\right){cos}^{k}\left(u\right){sin}^{m-k}\left(u\right)cos\left[\frac{\pi}{2}(m-k)\right]$$$$\begin{array}{c}\hfill N\left(u\right)=\sum _{k=0}^{m}\left(\genfrac{}{}{0pt}{}{m}{k}\right){cos}^{k}\left(u\right){sin}^{m-k}\left(u\right)sin\left[\frac{\pi}{2}(m-k)\right]=\\ \hfill \sum _{k=0}^{m}\left(\genfrac{}{}{0pt}{}{m}{k}\right){\left[\frac{{(1-{F}^{2})}^{2}-4{F}^{2}}{{(1+{F}^{2})}^{2}}\right]}^{k}{\left[\frac{4F(1-{F}^{2})}{{(1+{F}^{2})}^{2}}\right]}^{m-k}sin\left[\frac{\pi}{2}(m-k)\right].\end{array}$$$$\begin{array}{c}\hfill N\left(u\right)=\sum _{k=0}^{m}\left(\genfrac{}{}{0pt}{}{m}{k}\right){cos}^{k}\left(u\right){sin}^{m-k}\left(u\right)sin\left[\frac{\pi}{2}(m-k)\right]=\\ \hfill \sum _{k=0}^{m}\left(\genfrac{}{}{0pt}{}{m}{k}\right){\left[\frac{{(1-{F}^{2})}^{2}-4{F}^{2}}{{(1+{F}^{2})}^{2}}\right]}^{k}{\left[\frac{4F(1-{F}^{2})}{{(1+{F}^{2})}^{2}}\right]}^{m-k}cos\left[\frac{\pi}{2}(m-k)\right].\end{array}$$

## 4. Two Illustrative Examples

#### 4.1. Example 1

#### 4.2. Example 2

- $\overrightarrow{a}=({a}_{0},{a}_{1},\cdots ,{a}_{m})$;
- k: order of derivative of g;
- l: degree of derivative in the defining ODE;
- m: highest degree of the polynomial of g in the defining ODE.

- $\mathrm{F}=\mathrm{sn}(\mathrm{x};\mathrm{k})$: ${F}_{\xi}^{2}=(1-{F}^{2})(1-{k}^{2}{F}^{2})$;
- $\mathrm{F}=\mathrm{cn}(\mathrm{x};\mathrm{k})$: ${F}_{\xi}^{2}=(1-{F}^{2})({k}^{\prime 2}+{k}^{2}{F}^{2})$;
- $\mathrm{F}=\mathrm{dn}(\mathrm{x};\mathrm{k})$: ${F}_{\xi}^{2}=(1-{F}^{2})({F}^{2}-{k}^{\prime 2})$;

## 5. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Appendix A. Faa di Bruno Relationship for Derivatives of a Composite Function

- $\overrightarrow{\nu}=({\nu}_{1},\cdots ,{\nu}_{d})$ is a d-dimensional index containing the integer non-negative numbers ${\nu}_{1},\cdots ,{\nu}_{d}$;
- $\overrightarrow{z}=({z}_{1},\cdots ,{z}_{d})$ is a d-dimensional object containing the real numbers ${z}_{1},\cdots ,{z}_{d}$;
- $\mid \overrightarrow{\nu}\mid ={\displaystyle \sum _{i=1}^{d}}{\nu}_{i}$ is the sum of the elements of the d-dimensional index $\overrightarrow{\nu}$;
- $\overrightarrow{\nu}!={\displaystyle \prod _{i=1}^{d}}{\nu}_{i}!$ is the factorial of the multi-component index $\overrightarrow{\nu}$;
- ${\overrightarrow{z}}^{\overrightarrow{\nu}}={\displaystyle \prod _{i=1}^{d}}{z}_{i}^{{\nu}_{i}}$ is the $\overrightarrow{\nu}$-th power of the multi-component variable $\overrightarrow{z}$;
- ${D}_{\overrightarrow{x}}^{\overrightarrow{\nu}}=\frac{{\partial}^{\mid \overrightarrow{\nu}\mid}}{\partial {x}_{1}^{{\nu}_{1}}\cdots \partial {x}_{d}^{{\nu}_{d}}}$, $\mid \overrightarrow{\nu}\mid >0$ is the $\overrightarrow{\nu}$-th derivative with respect to the multi-component variable $\overrightarrow{x}$. We note that in this notation ${D}_{\overrightarrow{x}}^{\overrightarrow{0}}$ is the identity operator;
- $\mid \mid \overrightarrow{z}\mid \mid ={\displaystyle \underset{1\le i\le d}{max}}\mid {z}_{i}\mid $ is the maximum value component of the multi-component variable $\overrightarrow{z}$;
- For the d-dimensional index $\overrightarrow{l}=({l}_{1},\cdots ,{l}_{d})$ (${l}_{1},\cdots ,{l}_{d}$ are integers), we have $\overrightarrow{l}\le \overrightarrow{\nu}$ when ${l}_{i}\le {\nu}_{i},i=1,\cdots ,d$. Then we define$$\left(\genfrac{}{}{0pt}{}{\overrightarrow{\nu}}{\overrightarrow{l}}\right)=\prod _{i=1}^{d}\left(\genfrac{}{}{0pt}{}{{\nu}_{i}}{{l}_{i}}\right)=\frac{\overrightarrow{\nu}!}{\overrightarrow{l}!(\overrightarrow{\nu}-\overrightarrow{l})!}.$$
- Ordering of vector indices. For two vector indices, $\overrightarrow{\mu}=({\mu}_{1},\cdots ,{\mu}_{d})$ and $\overrightarrow{\nu}=({\nu}_{1},\cdots ,{\nu}_{d})$, we have $\overrightarrow{\mu}\prec \overrightarrow{\nu}$ when one of the following holds:
**(a.)**$\mid \overrightarrow{\mu}\mid <\mid \overrightarrow{\nu}\mid $;**(b.)**$\mid \overrightarrow{\mu}\mid =\mid \overrightarrow{\nu}\mid $ and ${\mu}_{1}<{\nu}_{1}$;**(c.)**$\mid \overrightarrow{\mu}\mid =\mid \overrightarrow{\nu}\mid $, ${\mu}_{1}={\nu}_{1}$, ⋯${\mu}_{k}={\nu}_{k}$ and ${\mu}_{k+1}<{\nu}_{k+1}$ for some $1\le k<d$.

- ${h}_{\left(n\right)}=\frac{{d}^{n}h}{d{x}^{n}}$ is the n-th derivative of the function h.
- ${f}_{\left(k\right)}=\frac{{d}^{k}f}{d{g}^{k}}$ is the k-th derivative of the function f.
- ${g}_{\left(i\right)}=\frac{{d}^{i}g}{d{x}^{i}}$ is the i-th derivative of the function g.
- $p(n,k)=\{{\lambda}_{1},{\lambda}_{2},\cdots ,{\lambda}_{n}\}$: set of numbers such that$$\sum _{i=1}^{n}{\lambda}_{i}=k;\sum _{i=1}^{n}i{\lambda}_{i}=n.$$

## Appendix B. Several Results Relevant for Applications of the SEsM in the Main Text

**(a.)**k: order of derivative of g;**(b.)**l: degree of derivative in the defining ODE;**(c.)**m: highest degree of the polynomial of g in the defining ODE.

**Theorem**

**A1.**

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**Figure 1.**The Simple Equations Method (SEsM) for the specific case of one solved equation by use of M simple equations. The method has four steps which are described in the text. The discussion in the text below is about the kinds of possible transformations used in Step (1.) of SEsM.

**Figure 2.**Examples of a kink and anti-kink described by the solution (14). The parameters are as follows: $t=0.2$. $v=0.2$, $A=3$, $l=1.3$, $b=1$, $d=2.2$$\alpha =0.08$, $\gamma =2.1$, $\delta =1.0d0$. ${\delta}_{1}=1$ for (

**a**); ${\delta}_{1}=-1$ for (

**b**).

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**MDPI and ACS Style**

Vitanov, N.K.; Dimitrova, Z.I.
Simple Equations Method and Non-Linear Differential Equations with Non-Polynomial Non-Linearity. *Entropy* **2021**, *23*, 1624.
https://doi.org/10.3390/e23121624

**AMA Style**

Vitanov NK, Dimitrova ZI.
Simple Equations Method and Non-Linear Differential Equations with Non-Polynomial Non-Linearity. *Entropy*. 2021; 23(12):1624.
https://doi.org/10.3390/e23121624

**Chicago/Turabian Style**

Vitanov, Nikolay K., and Zlatinka I. Dimitrova.
2021. "Simple Equations Method and Non-Linear Differential Equations with Non-Polynomial Non-Linearity" *Entropy* 23, no. 12: 1624.
https://doi.org/10.3390/e23121624