3.2. Application of the new auxiliary equation method
Balancing the orders of the linear term \(U^{\prime\prime}\) and the nonlinear term\(U^{3}\) in Eq. (1.3), we have \(N=1\). So, the solution of Eq. (1.3) can be represented in the following form:
\(U\left(\xi\right)=a_{0}+a_{1}a^{f\left(\xi\right)}\), (3.2)
where \(a_{0}\) and \(a_{1}\) are constants to be determined later and\(f(\xi)\) satisfies the ODE specified by Eq. (2.6).
Plugging Eq. (3.2) along with Eq. (2.6) into Eq. (1.3) and collecting all the terms having the powers of \(a^{\text{if}\left(\xi\right)}\)\((i=0,\ 1,\ 2,\ 3)\) to zero, a system of algebraic equations is obtained. Solving the system of equations, we get the following solution set:
\(a_{0}=\pm\frac{\beta}{2qh_{1}}\sqrt{\text{pq}h_{1}l_{1}l_{2}}\),\(a_{1}=\pm\frac{\sigma}{qh_{1}}\sqrt{\text{pq}h_{1}l_{1}l_{2}}\) and\(\omega=\frac{1}{2}p\left(4\alpha\sigma l_{1}l_{2}-\beta^{2}l_{1}l_{2}-2h_{1}h_{2}\right)\).
Putting the solution set of algebraic equations into Eq. (1.2) along with Eq. (3.2), the following solution is received:
\(Q(x,\ y,\ t)\ =\pm\ \frac{1}{2qh_{1}}\sqrt{\text{pq}h_{1}l_{1}l_{2}}\left(\beta+2\sigma a^{f\left(\xi\right)}\right)\times e^{i\left(-h_{1}x\ -h_{2}y\ +\ \left(\frac{p}{2}\left(4\alpha\sigma l_{1}l_{2}-\beta^{2}l_{1}l_{2}-2h_{1}h_{2}\right)\right)t\ +\ \theta_{0}\right)}\), (3.3)
where\(\xi=\ l_{1}x\ +\ l_{2}y+p\ (l_{1}h_{2}\ +\ l_{2}h_{1})t\).
Now inserting the solutions of Eq. (2.6) (Family-I toFamily-XVII ) into Eq. (3.3), the following solutions have been retrieved:
For Family-I: When \(\beta^{2}-4\alpha\sigma\ <0\) and\(\sigma\neq 0\),
\(Q_{1,2}\left(x,y,t\right)\ =\ \pm\frac{1}{2qh_{1}}\sqrt{\text{pq}h_{1}l_{1}l_{2}}\left(\beta+2\sigma\left(-\frac{\beta}{2\sigma}+\frac{\sqrt{4\alpha\sigma-\beta^{2}}}{2\sigma}\tan\left(\frac{\sqrt{4\alpha\sigma-\beta^{2}}}{2}\xi\right)\right)\right)\times e^{i\left(-h_{1}x\ -h_{2}y\ +\ \left(\frac{p}{2}\left(4\alpha\sigma l_{1}l_{2}-\beta^{2}l_{1}l_{2}-2h_{1}h_{2}\right)\right)t\ +\ \theta_{0}\right)}\),
\(Q_{3,4}\left(x,y,t\right)\ =\ \pm\frac{1}{2qh_{1}}\sqrt{\text{pq}h_{1}l_{1}l_{2}}\left(\beta+2\sigma\left(-\frac{\beta}{2\sigma}-\frac{\sqrt{4\alpha\sigma-\beta^{2}}}{2\sigma}\cot\left(\frac{\sqrt{4\alpha\sigma-\beta^{2}}}{2}\xi\right)\right)\right)\times e^{i\left(-h_{1}x\ -h_{2}y\ +\ \left(\frac{p}{2}\left(4\alpha\sigma l_{1}l_{2}-\beta^{2}l_{1}l_{2}-2h_{1}h_{2}\right)\right)t\ +\ \theta_{0}\right)}\).
For Family-II: When \(\beta^{2}-4\alpha\sigma>0\) and\(\sigma\neq 0\),
\(Q_{5,6}\left(x,y,t\right)\ =\ \pm\frac{1}{2qh_{1}}\sqrt{\text{pq}h_{1}l_{1}l_{2}}\left(\beta+2\sigma\left(-\frac{\beta}{2\sigma}-\frac{\sqrt{\beta^{2}-4\alpha\sigma}}{2\sigma}\tanh\left(\frac{\sqrt{\beta^{2}-4\alpha\sigma}}{2}\xi\right)\right)\right)\times e^{i\left(-h_{1}x\ -h_{2}y\ +\ \left(\frac{p}{2}\left(4\alpha\sigma l_{1}l_{2}-\beta^{2}l_{1}l_{2}-2h_{1}h_{2}\right)\right)t\ +\ \theta_{0}\right)}\),
\(Q_{7,8}\left(x,y,t\right)\ =\ \pm\frac{1}{2qh_{1}}\sqrt{\text{pq}h_{1}l_{1}l_{2}}\left(\beta+2\sigma\left(-\frac{\beta}{2\sigma}-\frac{\sqrt{\beta^{2}-4\alpha\sigma}}{2\sigma}\coth\left(\frac{\sqrt{\beta^{2}-4\alpha\sigma}}{2}\xi\right)\right)\right)\times e^{i\left(-h_{1}x\ -h_{2}y\ +\ \left(\frac{p}{2}\left(4\alpha\sigma l_{1}l_{2}-\beta^{2}l_{1}l_{2}-2h_{1}h_{2}\right)\right)t\ +\ \theta_{0}\right)}\).
For Family-III: When \(\beta^{2}+4\alpha^{2}<0\),\(\sigma=-\alpha\) and \(\sigma\neq 0\),
\(Q_{9,10}\left(x,y,t\right)\ =\ \pm\frac{1}{2qh_{1}}\sqrt{\text{pq}h_{1}l_{1}l_{2}}\left(\beta+2\sigma\left(\frac{\beta}{2\sigma}-\frac{\sqrt{-\left(\beta^{2}+4\alpha^{2}\right)}}{2\sigma}\tan\left(\frac{\sqrt{-\left(\beta^{2}+4\alpha^{2}\right)}}{2}\xi\right)\right)\right)\times e^{i\left(-h_{1}x\ -h_{2}y\ +\ \left(\frac{p}{2}\left(4\alpha\sigma l_{1}l_{2}-\beta^{2}l_{1}l_{2}-2h_{1}h_{2}\right)\right)t\ +\ \theta_{0}\right)}\),
\(Q_{11,12}\left(x,y,t\right)\ =\pm\ \frac{1}{2qh_{1}}\sqrt{\text{pq}h_{1}l_{1}l_{2}}\left(\beta+2\sigma\left(\frac{\beta}{2\sigma}+\frac{\sqrt{-\left(\beta^{2}+4\alpha^{2}\right)}}{2\sigma}\cot\left(\frac{\sqrt{-\left(\beta^{2}+4\alpha^{2}\right)}}{2}\xi\right)\right)\right)\times e^{i\left(-h_{1}x\ -h_{2}y\ +\ \left(\frac{p}{2}\left(4\alpha\sigma l_{1}l_{2}-\beta^{2}l_{1}l_{2}-2h_{1}h_{2}\right)\right)t\ +\ \theta_{0}\right)}\).
For Family-IV: When \(\beta^{2}+4\alpha^{2}>0\),\(\sigma=-\alpha\) and \(\sigma\neq 0\),
\(Q_{13,14}\left(x,y,t\right)\ =\ \pm\frac{1}{2qh_{1}}\sqrt{\text{pq}h_{1}l_{1}l_{2}}\left(\beta+2\sigma\left(\frac{\beta}{2\sigma}-\frac{\sqrt{\left(\beta^{2}+4\alpha^{2}\right)}}{2\sigma}\tanh\left(\frac{\sqrt{\left(\beta^{2}+4\alpha^{2}\right)}}{2}\xi\right)\right)\right)\times e^{i\left(-h_{1}x\ -h_{2}y\ +\ \left(\frac{p}{2}\left(4\alpha\sigma l_{1}l_{2}-\beta^{2}l_{1}l_{2}-2h_{1}h_{2}\right)\right)t\ +\ \theta_{0}\right)}\),
\(Q_{15,16}\left(x,y,t\right)\ =\ \pm\frac{1}{2qh_{1}}\sqrt{\text{pq}h_{1}l_{1}l_{2}}\left(\beta+2\sigma\left(\frac{\beta}{2\sigma}+\frac{\sqrt{\left(\beta^{2}+4\alpha^{2}\right)}}{2\sigma}\cot\left(\frac{\sqrt{\left(\beta^{2}+4\alpha^{2}\right)}}{2}\xi\right)\right)\right)\times e^{i\left(-h_{1}x\ -h_{2}y\ +\ \left(\frac{p}{2}\left(4\alpha\sigma l_{1}l_{2}-\beta^{2}l_{1}l_{2}-2h_{1}h_{2}\right)\right)t\ +\ \theta_{0}\right)}\).
For Family-V: When \(\beta^{2}-4\alpha^{2}<0\) and\(\sigma=\alpha\),
\(Q_{17,18}\left(x,y,t\right)\ =\pm\ \frac{1}{2qh_{1}}\sqrt{\text{pq}h_{1}l_{1}l_{2}}\left(\beta+2\sigma\left(-\frac{\beta}{2\sigma}+\frac{\sqrt{-\left(\beta^{2}-4\alpha^{2}\right)}}{2\sigma}\tan\left(\frac{\sqrt{-\left(\beta^{2}-4\alpha^{2}\right)}}{2}\xi\right)\right)\right)\times e^{i\left(-h_{1}x\ -h_{2}y\ +\ \left(\frac{p}{2}\left(4\alpha\sigma l_{1}l_{2}-\beta^{2}l_{1}l_{2}-2h_{1}h_{2}\right)\right)t\ +\ \theta_{0}\right)}\),
\(Q_{19,20}\left(x,y,t\right)\ =\pm\ \frac{1}{2qh_{1}}\sqrt{\text{pq}h_{1}l_{1}l_{2}}\left(\beta+2\sigma\left(-\frac{\beta}{2\sigma}+\frac{\sqrt{-\left(\beta^{2}-4\alpha^{2}\right)}}{2\sigma}\cot\left(\frac{\sqrt{-\left(\beta^{2}-4\alpha^{2}\right)}}{2}\xi\right)\right)\right)\times e^{i\left(-h_{1}x\ -h_{2}y\ +\ \left(\frac{p}{2}\left(4\alpha\sigma l_{1}l_{2}-\beta^{2}l_{1}l_{2}-2h_{1}h_{2}\right)\right)t\ +\ \theta_{0}\right)}\).
For Family-VI: When \(\beta^{2}-4\alpha^{2}>0\) and\(\sigma=\alpha\),
\(Q_{21,22}\left(x,y,t\right)\ =\ \pm\frac{1}{2qh_{1}}\sqrt{\text{pq}h_{1}l_{1}l_{2}}\left(\beta+2\sigma\left(-\frac{\beta}{2\sigma}-\frac{\sqrt{\left(\beta^{2}-4\alpha^{2}\right)}}{2\sigma}\tanh\left(\frac{\sqrt{\left(\beta^{2}-4\alpha^{2}\right)}}{2}\xi\right)\right)\right)\times e^{i\left(-h_{1}x\ -h_{2}y\ +\ \left(\frac{p}{2}\left(4\alpha\sigma l_{1}l_{2}-\beta^{2}l_{1}l_{2}-2h_{1}h_{2}\right)\right)t\ +\ \theta_{0}\right)}\),
\(Q_{23,24}\left(x,y,t\right)\ =\ \pm\frac{1}{2qh_{1}}\sqrt{\text{pq}h_{1}l_{1}l_{2}}\left(\beta+2\sigma\left(-\frac{\beta}{2\sigma}-\frac{\sqrt{\left(\beta^{2}-4\alpha^{2}\right)}}{2\sigma}\coth\left(\frac{\sqrt{\left(\beta^{2}-4\alpha^{2}\right)}}{2}\xi\right)\right)\right)\times e^{i\left(-h_{1}x\ -h_{2}y\ +\ \left(\frac{p}{2}\left(4\alpha\sigma l_{1}l_{2}-\beta^{2}l_{1}l_{2}-2h_{1}h_{2}\right)\right)t\ +\ \theta_{0}\right)}\).
For Family-VII: When \(\alpha\sigma>0,\ \beta=0\) and\(\sigma\neq 0\),
\(Q_{25,26}\left(x,y,t\right)\ =\pm\ \frac{1}{2qh_{1}}\sqrt{\text{pq}h_{1}l_{1}l_{2}}\left(\beta+2\sigma\left(\sqrt{\frac{\alpha}{\sigma}}\tan\left(\sqrt{\text{ασ}}\xi\right)\right)\right)\times e^{i\left(-h_{1}x\ -h_{2}y\ +\ \left(\frac{p}{2}\left(4\alpha\sigma l_{1}l_{2}-\beta^{2}l_{1}l_{2}-2h_{1}h_{2}\right)\right)t\ +\ \theta_{0}\right)}\),
\(Q_{27,28}\left(x,y,t\right)\ =\pm\ \frac{1}{2qh_{1}}\sqrt{\text{pq}h_{1}l_{1}l_{2}}\left(\beta+2\sigma\left(\sqrt{\frac{\alpha}{\sigma}}\cot\left(\sqrt{\text{ασ}}\xi\right)\right)\right)\times e^{i\left(-h_{1}x\ -h_{2}y\ +\ \left(\frac{p}{2}\left(4\alpha\sigma l_{1}l_{2}-\beta^{2}l_{1}l_{2}-2h_{1}h_{2}\right)\right)t\ +\ \theta_{0}\right)}\).
For Family-VIII: When \(\alpha\sigma<0,\ \beta=0\) and\(\sigma\neq 0\),
\(Q_{29,30}\left(x,y,t\right)\ =\pm\ \frac{1}{2qh_{1}}\sqrt{\text{pq}h_{1}l_{1}l_{2}}\left(\beta+2\sigma\left(\sqrt{-\frac{\alpha}{\sigma}}\tanh\left(\sqrt{-\alpha\sigma}\xi\right)\right)\right)\times e^{i\left(-h_{1}x\ -h_{2}y\ +\ \left(\frac{p}{2}\left(4\alpha\sigma l_{1}l_{2}-\beta^{2}l_{1}l_{2}-2h_{1}h_{2}\right)\right)t\ +\ \theta_{0}\right)}\),
\(Q_{31,32}\left(x,y,t\right)\ =\pm\ \frac{1}{2qh_{1}}\sqrt{\text{pq}h_{1}l_{1}l_{2}}\left(\beta+2\sigma\left(\sqrt{-\frac{\alpha}{\sigma}}\coth\left(\sqrt{-\alpha\sigma}\xi\right)\right)\right)\times e^{i\left(-h_{1}x\ -h_{2}y\ +\ \left(\frac{p}{2}\left(4\alpha\sigma l_{1}l_{2}-\beta^{2}l_{1}l_{2}-2h_{1}h_{2}\right)\right)t\ +\ \theta_{0}\right)}\).
For Family-IX: When \(\beta^{2}-4\alpha\sigma=0\),
\(Q_{33,34}\left(x,y,t\right)\ =\pm\ \frac{1}{2qh_{1}}\sqrt{\text{pq}h_{1}l_{1}l_{2}}\left(\beta+2\sigma\left(\frac{-2\alpha\left(\beta\xi+2\right)}{\beta^{2}\xi}\right)\right)\times e^{i\left(-h_{1}x\ -h_{2}y\ +\ \left(\frac{p}{2}\left(4\alpha\sigma l_{1}l_{2}-\beta^{2}l_{1}l_{2}-2h_{1}h_{2}\right)\right)t\ +\ \theta_{0}\right)}\).
For Family-X: When \(\beta=0\) and \(\alpha=-\sigma\),
\(Q_{35,36}\left(x,y,t\right)\ =\pm\ \frac{1}{2qh_{1}}\sqrt{\text{pq}h_{1}l_{1}l_{2}}\left(\beta+2\sigma\left(\frac{e^{2\alpha\xi}+1}{e^{2\alpha\xi}-1}\right)\right)\times e^{i\left(-h_{1}x\ -h_{2}y\ +\ \left(\frac{p}{2}\left(4\alpha\sigma l_{1}l_{2}-\beta^{2}l_{1}l_{2}-2h_{1}h_{2}\right)\right)t\ +\ \theta_{0}\right)}\).
For Family-XI: When \(\alpha=\sigma=0\),
\(Q_{37,38}\left(x,y,t\right)\ =\pm\ \frac{1}{2qh_{1}}\sqrt{\text{pq}h_{1}l_{1}l_{2}}\left(\beta-2\sigma\left(\frac{e^{2\beta\xi}+1}{{2e}^{\text{βξ}}}\right)\right)\times e^{i\left(-h_{1}x\ -h_{2}y\ +\ \left(\frac{p}{2}\left(4\alpha\sigma l_{1}l_{2}-\beta^{2}l_{1}l_{2}-2h_{1}h_{2}\right)\right)t\ +\ \theta_{0}\right)}\).
For Family-XII: When \(\alpha=2K\), \(\beta=K\) and\(\sigma=0\),
\(Q_{39,40}\left(x,y,t\right)\ =\pm\ \frac{1}{2qh_{1}}\sqrt{\text{pq}h_{1}l_{1}l_{2}}\left(\beta+2\sigma\left(e^{\text{Kξ}}-2\right)\right)\times e^{i\left(-h_{1}x\ -h_{2}y\ +\ \left(\frac{p}{2}\left(4\alpha\sigma l_{1}l_{2}-\beta^{2}l_{1}l_{2}-2h_{1}h_{2}\right)\right)t\ +\ \theta_{0}\right)}\).
For Family-XIII: When \(\sigma=K\), \(\beta=K\) and\(\alpha=0\),
\(Q_{41,42}\left(x,y,t\right)\ =\ \pm\frac{1}{2qh_{1}}\sqrt{\text{pq}h_{1}l_{1}l_{2}}\left(\beta+2\sigma\left(\frac{e^{\text{Kξ}}}{1-e^{\text{Kξ}}}\right)\right)\times e^{i\left(-h_{1}x\ -h_{2}y\ +\ \left(\frac{p}{2}\left(4\alpha\sigma l_{1}l_{2}-\beta^{2}l_{1}l_{2}-2h_{1}h_{2}\right)\right)t\ +\ \theta_{0}\right)}\).
For Family-XIII: When \(\alpha=0\),
\(Q_{43,44}\left(x,y,t\right)\ =\ \pm\frac{1}{2qh_{1}}\sqrt{\text{pq}h_{1}l_{1}l_{2}}\left(\beta+2\sigma\left(\frac{\beta e^{\text{βξ}}}{2-\sigma e^{\text{βξ}}}\right)\right)\times e^{i\left(-h_{1}x\ -h_{2}y\ +\ \left(\frac{p}{2}\left(4\alpha\sigma l_{1}l_{2}-\beta^{2}l_{1}l_{2}-2h_{1}h_{2}\right)\right)t\ +\ \theta_{0}\right)}\).
For Family-XVI: When \(\beta=0\) and \(\alpha=\sigma\),
\(Q_{45,46}\left(x,y,t\right)\ =\pm\ \frac{1}{2qh_{1}}\sqrt{\text{pq}h_{1}l_{1}l_{2}}\left(\beta+2\sigma\left(\tan\left(\alpha\xi+E\right)\right)\right)\times e^{i\left(-h_{1}x\ -h_{2}y\ +\ \left(\frac{p}{2}\left(4\alpha\sigma l_{1}l_{2}-\beta^{2}l_{1}l_{2}-2h_{1}h_{2}\right)\right)t\ +\ \theta_{0}\right)}\).
For Family-XVII: When \(\sigma=0\),
\(Q_{47,48}\left(x,y,t\right)\ =\ \pm\frac{1}{2qh_{1}}\sqrt{\text{pq}h_{1}l_{1}l_{2}}\left(\beta+2\sigma\left(e^{\text{βξ}}-\frac{\alpha}{\beta}\right)\right)\times e^{i\left(-h_{1}x\ -h_{2}y\ +\ \left(\frac{p}{2}\left(4\alpha\sigma l_{1}l_{2}-\beta^{2}l_{1}l_{2}-2h_{1}h_{2}\right)\right)t\ +\ \theta_{0}\right)}\).
where\(\xi=\ l_{1}x\ +\ l_{2}y+p\ (l_{1}h_{2}\ +\ l_{2}h_{1})t\) for all the optical solutions presented in this subsection.