2.1. Overview of the generalized Kudryashov method (gKM)
The main steps of the gKM are as follows [16]:
Let us assume that the solution \(U\left(\xi\right)\) of the nonlinear Eq. (2.2) can be presented as
\(U\left(\xi\right)=\frac{\sum_{i=0}^{N}{a_{i}R^{i}\left(\xi\right)}}{\sum_{j=0}^{M}{b_{j}R^{j}\left(\xi\right)}}\). (2.3)
In Eq. (2.3), the constants\(a_{i}\ \left(i=0,\ 1,\ 2\ldots,N\right)\) and\(b_{j}\ \left(j=0,\ 1,\ 2\ldots,M\right)\) are to be determined later. \(M\) and \(N\) are positive integers which can be computed by means of homogenous balance principle, and \(R(\xi)\) satisfies the following ODE:
\(R^{{}^{\prime}}\left(\xi\right)=\left(R^{2}\left(\xi\right)-R\left(\xi\right)\right)\text{\ ln}\left(a\right)\). (2.4)
Equation (2.4) yields the exact solution\(R\left(\xi\right)=\frac{1}{\left(1+da^{\xi}\right)}\), where\(a\neq 0,\ 1\) and \(d\neq 0\).
Substituting Eq. (2.3) along with Eq. (2.4) into Eq. (2.2) and using some mathematical operations, we get a system of algebraic equations. Solving the attained system and setting the obtained values in Eq. (2.3), one can produce exact solutions of Eq. (2.1).
2.2. Overview of the new auxiliary equation method (NAEM)
The main outlines of the NAEM are as follows [29]:
Let us assume that the solution \(U\left(\xi\right)\) of the nonlinear Eq. (2.2) can be presented as
\(U\left(\xi\right)=\sum_{i=0}^{N}{a_{i}\left(a^{f\left(\xi\right)}\right)^{i}}\), (2.5)
where the constant coefficients\(a_{i}\ \left(i=0,\ 1,\ 2\ldots,N\right)\) are to be determined and\(N\) is a positive integer which can be computed by means of homogenous balance principle. For Eq. (2.5), \(f(\xi)\) satisfies the following ODE:
\(f^{{}^{\prime}}\left(\xi\right)=\frac{1}{\ln\left(a\right)}\left(\alpha a^{-f\left(\xi\right)}+\beta+\sigma a^{f\left(\xi\right)}\right)\). (2.6)
Equation (2.6) yields the following family of solutions:
Family-I: When \(\beta^{2}-4\alpha\sigma\ <0\) and\(\sigma\neq 0\),
\(a^{f\left(\xi\right)}=-\frac{\beta}{2\sigma}+\frac{\sqrt{4\alpha\sigma-\beta^{2}}}{2\sigma}\tan\left(\frac{\sqrt{4\alpha\sigma-\beta^{2}}}{2}\xi\right)\),
\(a^{f\left(\xi\right)}=-\frac{\beta}{2\sigma}-\frac{\sqrt{4\alpha\sigma-\beta^{2}}}{2\sigma}\cot\left(\frac{\sqrt{4\alpha\sigma-\beta^{2}}}{2}\xi\right)\).
Family-II: When \(\beta^{2}-4\alpha\sigma>0\) and\(\sigma\neq 0\),
\(a^{f\left(\xi\right)}=-\frac{\beta}{2\sigma}-\frac{\sqrt{\beta^{2}-4\alpha\sigma}}{2\sigma}\tanh\left(\frac{\sqrt{\beta^{2}-4\alpha\sigma}}{2}\xi\right)\),
\(a^{f\left(\xi\right)}=-\frac{\beta}{2\sigma}-\frac{\sqrt{\beta^{2}-4\alpha\sigma}}{2\sigma}\coth\left(\frac{\sqrt{\beta^{2}-4\alpha\sigma}}{2}\xi\right)\).
Family-III: When \(\beta^{2}+4\alpha^{2}<0\),\(\sigma=-\alpha\) and \(\sigma\neq 0\),
\(a^{f\left(\xi\right)}=\frac{\beta}{2\sigma}-\frac{\sqrt{-\left(\beta^{2}+4\alpha^{2}\right)}}{2\sigma}\tan\left(\frac{\sqrt{-\left(\beta^{2}+4\alpha^{2}\right)}}{2}\xi\right)\),
\(a^{f\left(\xi\right)}=\frac{\beta}{2\sigma}+\frac{\sqrt{-\left(\beta^{2}+4\alpha^{2}\right)}}{2\sigma}\cot\left(\frac{\sqrt{-\left(\beta^{2}+4\alpha^{2}\right)}}{2}\xi\right)\).
Family-IV: When \(\beta^{2}+4\alpha^{2}>0\),\(\sigma=-\alpha\) and \(\sigma\neq 0\),
\(a^{f\left(\xi\right)}=\frac{\beta}{2\sigma}-\frac{\sqrt{\left(\beta^{2}+4\alpha^{2}\right)}}{2\sigma}\tanh\left(\frac{\sqrt{\left(\beta^{2}+4\alpha^{2}\right)}}{2}\xi\right)\),
\(a^{f\left(\xi\right)}=\frac{\beta}{2\sigma}+\frac{\sqrt{\left(\beta^{2}+4\alpha^{2}\right)}}{2\sigma}\cot\left(\frac{\sqrt{\left(\beta^{2}+4\alpha^{2}\right)}}{2}\xi\right)\).
Family-V: When \(\beta^{2}-4\alpha^{2}<0\) and\(\sigma=\alpha\),
\(a^{f\left(\xi\right)}=-\frac{\beta}{2\sigma}+\frac{\sqrt{-\left(\beta^{2}-4\alpha^{2}\right)}}{2\sigma}\tan\left(\frac{\sqrt{-\left(\beta^{2}-4\alpha^{2}\right)}}{2}\xi\right)\),
\(a^{f\left(\xi\right)}=-\frac{\beta}{2\sigma}+\frac{\sqrt{-\left(\beta^{2}-4\alpha^{2}\right)}}{2\sigma}\cot\left(\frac{\sqrt{-\left(\beta^{2}-4\alpha^{2}\right)}}{2}\xi\right)\).
Family-VI: When \(\beta^{2}-4\alpha^{2}>0\) and\(\sigma=\alpha\),
\(a^{f\left(\xi\right)}=-\frac{\beta}{2\sigma}-\frac{\sqrt{\left(\beta^{2}-4\alpha^{2}\right)}}{2\sigma}\tanh\left(\frac{\sqrt{\left(\beta^{2}-4\alpha^{2}\right)}}{2}\xi\right)\),
\(a^{f\left(\xi\right)}=-\frac{\beta}{2\sigma}-\frac{\sqrt{\left(\beta^{2}-4\alpha^{2}\right)}}{2\sigma}\coth\left(\frac{\sqrt{\left(\beta^{2}-4\alpha^{2}\right)}}{2}\xi\right)\).
Family-VII: When \(\alpha\sigma>0,\ \beta=0\) and\(\sigma\neq 0\),
\(a^{f\left(\xi\right)}=\sqrt{\frac{\alpha}{\sigma}}\tan\left(\sqrt{\text{ασ}}\xi\right)\),
\(a^{f\left(\xi\right)}=\sqrt{\frac{\alpha}{\sigma}}\cot\left(\sqrt{\text{ασ}}\xi\right)\).
Family-VIII: When \(\alpha\sigma<0,\ \beta=0\) and\(\sigma\neq 0\),
\(a^{f\left(\xi\right)}=\sqrt{-\frac{\alpha}{\sigma}}\tanh\left(\sqrt{-\alpha\sigma}\xi\right)\),
\(a^{f\left(\xi\right)}=\sqrt{-\frac{\alpha}{\sigma}}\coth\left(\sqrt{-\alpha\sigma}\xi\right)\).
Family-IX: When \(\beta^{2}-4\alpha\sigma=0\),
\(a^{f\left(\xi\right)}=\frac{-2\alpha\left(\beta\xi+2\right)}{\beta^{2}\xi}\).
Family-X: When \(\beta=0\) and \(\alpha=-\sigma\),
\(a^{f\left(\xi\right)}=\frac{e^{2\alpha\xi}+1}{e^{2\alpha\xi}-1}\).
Family-XI: When \(\alpha=\sigma=0\),
\(a^{f\left(\xi\right)}=-\frac{e^{2\beta\xi}+1}{{2e}^{\text{βξ}}}\).
Family-XII: When \(\alpha=2K\), \(\beta=K\) and\(\sigma=0\),
\(a^{f\left(\xi\right)}=e^{\text{Kξ}}-2\).
Family-XIII: When \(\sigma=K\), \(\beta=K\) and\(\alpha=0\),
\(a^{f\left(\xi\right)}=\frac{e^{\text{Kξ}}}{1-e^{\text{Kξ}}}\).
Family-XIII: When \(\alpha=0\),
\(a^{f\left(\xi\right)}=\frac{\beta e^{\text{βξ}}}{2-\sigma e^{\text{βξ}}}\).
Family-XIV: When \(\beta=\sigma=0\),
\(a^{f\left(\xi\right)}=\alpha\xi\).
Family-XV: When \(\beta=\alpha=0\),
\(a^{f\left(\xi\right)}=-\frac{1}{\text{σξ}}\).
Family-XVI: When \(\beta=0\) and \(\alpha=\sigma\),
\(a^{f\left(\xi\right)}=tan\left(\alpha\xi+E\right)\).
Family-XVII: When \(\sigma=0\),
\(a^{f\left(\xi\right)}=e^{\text{βξ}}-\frac{\alpha}{\beta}\).
Plugging Eqs. (2.5) and (2.6) into Eq. (2.2) and collecting all the terms having the powers of \(a^{\text{if}\left(\xi\right)}\)\((i=0,\ 1,\ 2,\ 3\ldots)\) to zero, a system of algebraic equations is obtained. This system can be probed to determine the values of\(a_{i},\ \omega\) and others in Eq. (2.2), which finally produce exact solutions to Eq. (2.1).