Equation (1) can be rewritten as follows:

$$\begin{array}{c}4{u}^{2}{u}_{t}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}4{u}^{2}{u}_{xxx}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}6u{u}_{x}{u}_{xx}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}3{u}_{x}^{3}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}8A{u}^{3}{v}_{x}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}8A{u}^{2}{u}_{x}v\text{\hspace{0.17em}}=\text{\hspace{0.17em}}0,\\ {u}_{x}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{v}_{y}.\end{array}\text{\hspace{1em}(2)}$$(2)

We assume that (2) has the following ansatz:

$$\begin{array}{c}u\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{a}_{0}\mathrm{(}x,\text{\hspace{0.17em}}y,\text{\hspace{0.17em}}t\mathrm{)}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{\displaystyle \sum _{i\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1}^{m}\{{a}_{i}\mathrm{(}x,\text{\hspace{0.17em}}y,\text{\hspace{0.17em}}t\mathrm{)}{\varphi}^{i}[R\mathrm{(}x,\text{\hspace{0.17em}}y,\text{\hspace{0.17em}}t\mathrm{)}]\text{\hspace{0.17em}}\begin{array}{c}\\ \end{array}}\\ +\text{\hspace{0.17em}}\frac{{b}_{i}\mathrm{(}x,\text{\hspace{0.17em}}y,\text{\hspace{0.17em}}t\mathrm{)}}{{\varphi}^{i}[R\mathrm{(}x,\text{\hspace{0.17em}}y,\text{\hspace{0.17em}}t\mathrm{)}]}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{c}_{i}\mathrm{(}x,\text{\hspace{0.17em}}y,\text{\hspace{0.17em}}t\mathrm{)}{\varphi}^{i\text{\hspace{0.17em}}-\text{\hspace{0.17em}}1}\\ \begin{array}{c}\\ \end{array}[R\mathrm{(}x,\text{\hspace{0.17em}}y,\text{\hspace{0.17em}}t\mathrm{)}]\sqrt{{\varphi}^{\prime}[R\mathrm{(}x,\text{\hspace{0.17em}}y,\text{\hspace{0.17em}}t\mathrm{)}]}\},\\ v\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{d}_{0}\mathrm{(}x,\text{\hspace{0.17em}}y,\text{\hspace{0.17em}}t\mathrm{)}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{\displaystyle \sum _{i\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1}^{n}\{{d}_{j}\mathrm{(}x,\text{\hspace{0.17em}}y,\text{\hspace{0.17em}}t\mathrm{)}{\varphi}^{j}[R\mathrm{(}x,\text{\hspace{0.17em}}y,\text{\hspace{0.17em}}t\mathrm{)}]\begin{array}{c}\\ \end{array}}\\ +\text{\hspace{0.17em}}\frac{{e}_{j}\mathrm{(}x,\text{\hspace{0.17em}}y,\text{\hspace{0.17em}}t\mathrm{)}}{{\varphi}^{j}[R\mathrm{(}x,\text{\hspace{0.17em}}y,\text{\hspace{0.17em}}t\mathrm{)}]}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{f}_{j}\mathrm{(}x,\text{\hspace{0.17em}}y,\text{\hspace{0.17em}}t\mathrm{)}{\varphi}^{j\text{\hspace{0.17em}}-\text{\hspace{0.17em}}1}\\ \begin{array}{c}\\ \end{array}[R\mathrm{(}x,\text{\hspace{0.17em}}y,\text{\hspace{0.17em}}t\mathrm{)}]\sqrt{{\varphi}^{\prime}[R\mathrm{(}x,\text{\hspace{0.17em}}y,\text{\hspace{0.17em}}t\mathrm{)}]}\},\end{array}\text{\hspace{1em}(3)}$$(3)

where *a*_{0}, *a*_{i}, *b*_{i}, *c*_{i}, *d*_{j}, *e*_{j}, and *f*_{j} are all functions of {*x*, *y*, *t*}, and *ϕ* is a solution of the Riccati equation

$${\varphi}^{\prime}\text{\hspace{0.17em}}\equiv \text{\hspace{0.17em}}\frac{\text{d}\varphi}{\text{d}R}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{l}_{0}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{l}_{1}\varphi \text{\hspace{0.17em}}+\text{\hspace{0.17em}}{l}_{2}{\varphi}^{2}\mathrm{,}\text{\hspace{1em}(4)}$$(4)

whose four types and 27 solutions were listed in [10], and here, we neglect them.

Balancing the highest-order partial differential terms and the highest nonlinear terms in (2), we have *m*=*n*=2. Substituting ansatz (3) with (4) into (2), selecting the variable separation form *R*=*p*(*x*, *t*)+*q*(*y*) and eliminating all the coefficients of polynomials of *ϕ* and $$\sqrt{{l}_{0}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{l}_{1}\varphi \mathrm{(}R\mathrm{)}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{l}_{2}{\varphi}^{2}\mathrm{(}R\mathrm{)}}$$ yields two families of solutions

**Family 1**

$$\begin{array}{l}{b}_{1}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{c}_{1}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{b}_{2}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{c}_{2}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{e}_{1}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{f}_{1}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{e}_{2}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{f}_{2}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}0,\text{\hspace{0.17em}}{a}_{0}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{l}_{0}{l}_{2}{p}_{x}{q}_{y}}{2A},\text{\hspace{0.17em}}\\ {a}_{1}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{l}_{1}{l}_{2}{p}_{x}{q}_{y}}{2A},\text{\hspace{0.17em}}{a}_{2}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{l}_{2}^{2}{p}_{x}{q}_{y}}{2A},\text{\hspace{0.17em}}\\ {d}_{0}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{6{l}_{1}{p}_{xx}{p}_{x}^{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}3{p}_{xx}^{2}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\mathrm{(}8{l}_{0}{l}_{2}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{l}_{1}^{2}\mathrm{)}{p}_{x}^{4}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}4{p}_{x}{p}_{t}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}4{p}_{x}{p}_{xxx}}{8A{p}_{x}^{2}},\text{\hspace{0.17em}}\\ {d}_{1}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{l}_{2}\mathrm{(}{l}_{1}{p}_{x}^{2}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{p}_{xx}\mathrm{)}}{2A},\text{\hspace{0.17em}}{d}_{2}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{l}_{2}^{2}{p}_{x}^{2}}{2A},\end{array}\text{\hspace{1em}(5)}$$(5)

where *p*≡ *p*(*x*, *t*) and *q*=*q*(*y*) are arbitrary functions of {*x*, *t*} and {*y*}, respectively.

**Family 2**

$$\begin{array}{l}{a}_{1}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{b}_{1}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{b}_{2}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{c}_{1}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{e}_{1}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{e}_{2}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}0,\text{\hspace{0.17em}}{a}_{0}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{l}_{0}{l}_{2}{p}_{x}{q}_{y}}{4A},\text{\hspace{0.17em}}{a}_{2}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{l}_{2}^{2}{p}_{x}{q}_{y}}{4A},\text{\hspace{0.17em}}\\ {c}_{2}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{l}_{2}^{3/2}{p}_{x}{q}_{y}}{4A},\text{\hspace{0.17em}}{d}_{0}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{12{p}_{xx}^{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}20{l}_{0}{l}_{2}{p}_{x}^{4}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}16{p}_{x}{p}_{t}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}16{p}_{x}{p}_{xxx}}{32A{p}_{x}^{2}},\text{\hspace{0.17em}}\\ {d}_{1}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{l}_{2}{p}_{xx}}{4A},\text{\hspace{0.17em}}{d}_{2}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{l}_{2}^{2}{p}_{x}^{2}}{4A},\text{\hspace{0.17em}}{f}_{1}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{l}_{2}^{1/2}{p}_{xx}}{4A},\text{\hspace{0.17em}}{f}_{2}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{l}_{2}^{3/2}{p}_{x}^{2}}{4A},\end{array}\text{\hspace{1em}(6)}$$(6)

with *l*_{1}=0, *p*≡ *p*(*x*, *t*), and *q*=*q*(*y*) are arbitrary functions of {*x*, *t*} and {*y*}, respectively.

**Family 3**

$$\begin{array}{l}{a}_{1}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{b}_{1}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{c}_{1}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{c}_{2}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{f}_{1}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{f}_{2}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}0,\text{\hspace{0.17em}}{a}_{0}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{l}_{0}{l}_{2}{p}_{x}{q}_{y}}{A},\text{\hspace{0.17em}}{a}_{2}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{l}_{2}^{2}{p}_{x}{q}_{y}}{2A},\\ {b}_{2}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{l}_{0}^{2}{p}_{x}{q}_{y}}{2A},\text{\hspace{0.17em}}{d}_{0}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{3{p}_{xx}^{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}8{l}_{0}{l}_{2}{p}_{x}^{4}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}4{p}_{x}{p}_{t}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}4{p}_{x}{p}_{xxx}}{8A{p}_{x}^{2}},\text{\hspace{0.17em}}\\ {d}_{1}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{l}_{2}{p}_{xx}}{2A},\text{\hspace{0.17em}}{d}_{2}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{l}_{2}^{2}{p}_{x}^{2}}{2A},\text{\hspace{0.17em}}{e}_{1}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{l}_{0}{p}_{xx}}{2A},\text{\hspace{0.17em}}{e}_{2}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{l}_{0}^{2}{p}_{x}^{2}}{2A},\end{array}\text{\hspace{1em}(7)}$$(7)

with *l*_{1}=0, *p*≡ *p*(*x*, *t*), and *q*=*q*(*y*) are arbitrary functions of {*x*, *t*} and {*y*}, respectively.

The variable separation solutions of the (2+1)-dimensional mKdV equation can be written as follows:

**Family 1**

From the above result (5) and some typical solutions of (4) in Zhu [10], we have

**Case 1:** For $${l}_{1}^{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}4{l}_{0}{l}_{2}\text{\hspace{0.17em}}>\text{\hspace{0.17em}}0$$ and *l*_{0}*l*_{1} ≠ 0 (or *l*_{0}*l*_{2} ≠ 0),

**Solution 1**

$$\begin{array}{c}{u}_{1}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{l}_{0}{l}_{2}{p}_{x}{q}_{y}}{2A}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\frac{3{l}_{1}{p}_{x}{q}_{y}}{4A}\left\{{l}_{1}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\sqrt{{l}_{1}^{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}4{l}_{0}{l}_{2}}\text{tanh}\left[\frac{\sqrt{{l}_{1}^{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}4{l}_{0}{l}_{2}}}{2}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}\right]\right\}\\ \text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{p}_{x}{q}_{y}}{8A}{\left\{{l}_{1}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\sqrt{{l}_{1}^{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}4{l}_{0}{l}_{2}}\text{tanh}\left[\frac{\sqrt{{l}_{1}^{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}4{l}_{0}{l}_{2}}}{2}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}\right]\right\}}^{2},\\ {v}_{1}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{6{l}_{1}{p}_{xx}{p}_{x}^{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}3{p}_{xx}^{2}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\mathrm{(}8{l}_{0}{l}_{2}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{l}_{1}^{2}\mathrm{)}{p}_{x}^{4}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}4{p}_{x}{p}_{t}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}4{p}_{x}{p}_{xxx}}{8A{p}_{x}^{2}}\text{\hspace{0.17em}}\\ \text{\hspace{0.17em}}+\text{\hspace{0.17em}}\frac{3\mathrm{(}{l}_{1}{p}_{x}^{2}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{p}_{xx}\mathrm{)}}{4A}\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}\left\{{l}_{1}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\sqrt{{l}_{1}^{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}4{l}_{0}{l}_{2}}\text{tanh}\left[\frac{\sqrt{{l}_{1}^{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}4{l}_{0}{l}_{2}}}{2}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}\right]\right\}\\ -\text{\hspace{0.17em}}\frac{3{p}_{x}^{2}}{8A}{\left\{{l}_{1}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\sqrt{{l}_{1}^{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}4{l}_{0}{l}_{2}}\text{tanh}\left[\frac{\sqrt{{l}_{1}^{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}4{l}_{0}{l}_{2}}}{2}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}\right]\right\}}^{2},\end{array}\text{\hspace{1em}(8)}$$(8)

**Solution 2**

$$\begin{array}{c}{u}_{2}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{l}_{0}{l}_{2}{p}_{x}{q}_{y}}{2A}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\frac{3{l}_{1}{p}_{x}{q}_{y}}{4A}\left\{{l}_{1}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\sqrt{{l}_{1}^{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}4{l}_{0}{l}_{2}}\text{coth}\left[\frac{\sqrt{{l}_{1}^{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}4{l}_{0}{l}_{2}}}{2}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}\right]\right\}\\ \text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{p}_{x}{q}_{y}}{8A}{\left\{{l}_{1}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\sqrt{{l}_{1}^{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}4{l}_{0}{l}_{2}}\text{coth}\left[\frac{\sqrt{{l}_{1}^{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}4{l}_{0}{l}_{2}}}{2}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}\right]\right\}}^{2},\\ {v}_{2}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{6{l}_{1}{p}_{xx}{p}_{x}^{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}3{p}_{xx}^{2}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\mathrm{(}8{l}_{0}{l}_{2}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{l}_{1}^{2}\mathrm{)}{p}_{x}^{4}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}4{p}_{x}{p}_{t}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}4{p}_{x}{p}_{xxx}}{8A{p}_{x}^{2}}\\ \text{\hspace{0.17em}}+\text{\hspace{0.17em}}\frac{3\mathrm{(}{l}_{1}{p}_{x}^{2}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{p}_{xx}\mathrm{)}}{4A}\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}\left\{{l}_{1}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\sqrt{{l}_{1}^{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}4{l}_{0}{l}_{2}}\text{coth}\left[\frac{\sqrt{{l}_{1}^{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}4{l}_{0}{l}_{2}}}{2}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}\right]\right\}\\ \text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{p}_{x}^{2}}{8A}{\left\{{l}_{1}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\sqrt{{l}_{1}^{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}4{l}_{0}{l}_{2}}\text{coth}\left[\frac{\sqrt{{l}_{1}^{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}4{l}_{0}{l}_{2}}}{2}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}\right]\right\}}^{2},\end{array}\text{\hspace{1em}(9)}$$(9)

**Case 2:** For $${l}_{1}^{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}4{l}_{0}{l}_{2}\text{\hspace{0.17em}}<\text{\hspace{0.17em}}0$$ and *l*_{0}*l*_{1} ≠ 0 (or *l*_{0}*l*_{2} ≠ 0),

**Solution 3**

$$\begin{array}{c}{u}_{3}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{l}_{0}{l}_{2}{p}_{x}{q}_{y}}{2A}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{l}_{1}{p}_{x}{q}_{y}}{4A}\left\{-\text{\hspace{0.17em}}{l}_{1}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\sqrt{4{l}_{0}{l}_{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{l}_{1}^{2}}\text{tan}\left[\frac{\sqrt{4{l}_{0}{l}_{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{l}_{1}^{2}}}{2}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}\right]\right\}\\ \text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{p}_{x}{q}_{y}}{8A}{\left\{-\text{\hspace{0.17em}}{l}_{1}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\sqrt{4{l}_{0}{l}_{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{l}_{1}^{2}}\text{tan}\left[\frac{\sqrt{4{l}_{0}{l}_{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{l}_{1}^{2}}}{2}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}\right]\right\}}^{2},\\ {v}_{3}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{6{l}_{1}{p}_{xx}{p}_{x}^{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}3{p}_{xx}^{2}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\mathrm{(}8{l}_{0}{l}_{2}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{l}_{1}^{2}\mathrm{)}{p}_{x}^{4}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}4{p}_{x}{p}_{t}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}4{p}_{x}{p}_{xxx}}{8A{p}_{x}^{2}}\\ \text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3\mathrm{(}{l}_{1}{p}_{x}^{2}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{p}_{xx}\mathrm{)}}{4A}\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}\left\{-{l}_{1}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\sqrt{4{l}_{0}{l}_{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{l}_{1}^{2}}\text{tan}\left[\frac{\sqrt{4{l}_{0}{l}_{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{l}_{1}^{2}}}{2}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}\right]\right\}\\ \text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{p}_{x}^{2}}{8A}{\left\{-\text{\hspace{0.17em}}{l}_{1}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\sqrt{4{l}_{0}{l}_{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{l}_{1}^{2}}\text{tan}\left[\frac{\sqrt{4{l}_{0}{l}_{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{l}_{1}^{2}}}{2}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}\right]\right\}}^{2},\end{array}\text{\hspace{1em}(10)}$$(10)

**Solution 4**

$$\begin{array}{c}{u}_{4}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{l}_{0}{l}_{2}{p}_{x}{q}_{y}}{2A}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\frac{3{l}_{1}{p}_{x}{q}_{y}}{4A}\left\{{l}_{1}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\sqrt{4{l}_{0}{l}_{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{l}_{1}^{2}}\mathrm{cot}\left[\frac{\sqrt{4{l}_{0}{l}_{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{l}_{1}^{2}}}{2}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}\right]\right\}\\ \text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{p}_{x}{q}_{y}}{8A}{\left\{{l}_{1}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\sqrt{4{l}_{0}{l}_{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{l}_{1}^{2}}\mathrm{cot}\left[\frac{\sqrt{4{l}_{0}{l}_{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{l}_{1}^{2}}}{2}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}\right]\right\}}^{2},\\ {v}_{4}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{6{l}_{1}{p}_{xx}{p}_{x}^{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}3{p}_{xx}^{2}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\mathrm{(}8{l}_{0}{l}_{2}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{l}_{1}^{2}\mathrm{)}{p}_{x}^{4}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}4{p}_{x}{p}_{t}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}4{p}_{x}{p}_{xxx}}{8A{p}_{x}^{2}}\\ \text{\hspace{0.17em}}+\text{\hspace{0.17em}}\frac{3\mathrm{(}{l}_{1}{p}_{x}^{2}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{p}_{xx}\mathrm{)}}{4A}\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}\left\{{l}_{1}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\sqrt{4{l}_{0}{l}_{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{l}_{1}^{2}}\mathrm{cot}\left[\frac{\sqrt{4{l}_{0}{l}_{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{l}_{1}^{2}}}{2}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}\right]\right\}\\ \text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{p}_{x}^{2}}{8A}{\left\{{l}_{1}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\sqrt{4{l}_{0}{l}_{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{l}_{1}^{2}}\mathrm{cot}\left[\frac{\sqrt{4{l}_{0}{l}_{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{l}_{1}^{2}}}{2}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}\right]\right\}}^{2},\end{array}\text{\hspace{1em}(11)}$$(11)

**Case 3:** For *l*_{2} ≠ 0 and *l*_{0}=*l*_{1}=0,

**Solution 5**

$$\begin{array}{c}{u}_{5}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{p}_{x}{q}_{y}}{2A{\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}}^{2}},\text{\hspace{0.17em}}{v}_{5}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{3{p}_{xx}^{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}4{p}_{x}{p}_{t}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}4{p}_{x}{p}_{xxx}}{8A{p}_{x}^{2}}\\ \text{\hspace{0.17em}}+\text{\hspace{0.17em}}\frac{3{p}_{xx}}{2A\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{p}_{x}^{2}}{2A{\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}}^{2}},\end{array}\text{\hspace{1em}(12)}$$(12)

where *p*≡ *p*(*x*, *t*) and *q*=*q*(*y*) are arbitrary functions of {*x*, *t*} and {*y*}, respectively.

**Family 2**

From the above result (6) and some typical solutions of (4) with *l*_{1}=0 in Dai and Wang [22], we have

**Case 1:** For *l*_{0}*l*_{2}=−1,

**Solution 6**

$$\begin{array}{c}{u}_{6}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{3{p}_{x}{q}_{y}}{4A}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{p}_{x}{q}_{y}}{4A}{\mathrm{tanh}}_{{l}_{0}}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}[{\mathrm{tanh}}_{{l}_{0}}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}\\ +\text{\hspace{0.17em}}{\text{isech}}_{{l}_{0}}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}],\\ {v}_{6}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{12{p}_{xx}^{2}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}20{p}_{x}^{4}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}16{p}_{x}{p}_{t}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}16{p}_{x}{p}_{xxx}}{32A{p}_{x}^{2}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\frac{3{p}_{xx}}{4A}[{\mathrm{tanh}}_{{l}_{0}}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}\\ +\text{\hspace{0.17em}}{\text{isech}}_{{l}_{0}}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}]\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{p}_{x}^{2}}{4A}{\mathrm{tanh}}_{{l}_{0}}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}[{\mathrm{tanh}}_{{l}_{0}}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}\\ +\text{\hspace{0.17em}}{\text{isech}}_{{l}_{0}}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}],\end{array}\text{\hspace{1em}(13)}$$(13)

**Solution 7**

$$\begin{array}{c}{u}_{7}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{3{p}_{x}{q}_{y}}{4A}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{p}_{x}{q}_{y}}{4A}{\mathrm{coth}}_{{l}_{0}}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}[{\mathrm{coth}}_{{l}_{0}}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}\\ -\text{\hspace{0.17em}}{\text{csch}}_{{l}_{0}}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}],\\ {v}_{7}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{12{p}_{xx}^{2}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}20{p}_{x}^{4}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}16{p}_{x}{p}_{t}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}16{p}_{x}{p}_{xxx}}{32A{p}_{x}^{2}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\frac{3{p}_{xx}}{4A}[{\mathrm{coth}}_{{l}_{0}}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}\\ -\text{\hspace{0.17em}}{\text{csch}}_{{l}_{0}}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}]\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{p}_{x}^{2}}{4A}{\mathrm{coth}}_{{l}_{0}}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}[{\mathrm{coth}}_{{l}_{0}}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}\\ \text{\hspace{0.17em}}-\text{\hspace{0.17em}}{\text{csch}}_{{l}_{0}}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}],\end{array}\text{\hspace{1em}(14)}$$(14)

**Case 2:** For *l*_{0}*l*_{1}=1,

**Solution 8**

$$\begin{array}{c}{u}_{8}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{p}_{x}{q}_{y}}{4A}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{p}_{x}{q}_{y}}{4A}{\mathrm{tan}}_{{l}_{0}}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}[{\mathrm{tan}}_{{l}_{0}}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}]\text{\hspace{0.17em}}\\ +\text{\hspace{0.17em}}{\mathrm{sec}}_{{l}_{0}}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}],\\ {v}_{8}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{12{p}_{xx}^{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}20{p}_{x}^{4}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}16{p}_{x}{p}_{t}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}16{p}_{x}{p}_{xxx}}{32A{p}_{x}^{2}}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{p}_{xx}}{4A}[{\mathrm{tan}}_{{l}_{0}}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}\\ +\text{\hspace{0.17em}}{\mathrm{sec}}_{{l}_{0}}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}]\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{p}_{x}^{2}}{4A}{\mathrm{tan}}_{{l}_{0}}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}[{\mathrm{tan}}_{{l}_{0}}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}\text{\hspace{0.17em}}\\ +\text{\hspace{0.17em}}{\mathrm{sec}}_{{l}_{0}}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}],\end{array}\text{\hspace{1em}(15)}$$(15)

**Solution 9**

$$\begin{array}{c}{u}_{9}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{p}_{x}{q}_{y}}{4A}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{p}_{x}{q}_{y}}{4A}{\mathrm{cot}}_{{l}_{0}}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}[{\mathrm{cot}}_{{l}_{0}}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{\mathrm{csc}}_{{l}_{0}}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}],\\ {v}_{9}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{12{p}_{xx}^{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}20{p}_{x}^{4}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}16{p}_{x}{p}_{t}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}16{p}_{x}{p}_{xxx}}{32A{p}_{x}^{2}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\frac{3{p}_{xx}}{4A}[{\mathrm{csc}}_{{l}_{0}}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}\\ -\text{\hspace{0.17em}}{\mathrm{cot}}_{{l}_{0}}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}]\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{p}_{x}^{2}}{4A}{\mathrm{cot}}_{{l}_{0}}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}[{\mathrm{csc}}_{{l}_{0}}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}\\ -\text{\hspace{0.17em}}{\mathrm{cot}}_{{l}_{0}}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}],\end{array}\text{\hspace{1em}(16)}$$(16)

where *p*≡ *p*(*x*, *t*) and *q*=*q*(*y*) are arbitrary functions of {*x*, *t*} and {*y*}, respectively. Moreover, functions in solutions (13)–(16) are *l*-deformed functions [23], whose properties will be recalled, that is,

$$\begin{array}{l}{\mathrm{sinh}}_{l}\mathrm{(}R\mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{{e}^{R}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}l{e}^{\text{\hspace{0.22em}}-R}}{2},\text{\hspace{0.17em}}{\mathrm{cosh}}_{l}\mathrm{(}R\mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{{e}^{R}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}l{e}^{\text{\hspace{0.22em}}-R}}{2},\\ {\mathrm{tanh}}_{l}\mathrm{(}R\mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{{\mathrm{sinh}}_{l}\mathrm{(}R\mathrm{)}}{{\mathrm{cosh}}_{l}\mathrm{(}R\mathrm{)}},{\text{\hspace{0.17em}sech}}_{l}\mathrm{(}R\mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{1}{{\mathrm{cosh}}_{l}\mathrm{(}R\mathrm{)}},\text{\hspace{0.17em}}R\text{\hspace{0.17em}}\in \text{\hspace{0.17em}}C.\end{array}\text{\hspace{1em}(17)}$$(17)

It is straightforward to see that the following formulas hold,

$$\begin{array}{l}\mathrm{(}{\mathrm{sinh}}_{l}\mathrm{(}R\mathrm{)}{\mathrm{)}}^{\prime}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\text{cosh}}_{l}\mathrm{(}R\mathrm{)},\text{\hspace{0.17em}}\mathrm{(}{\mathrm{cosh}}_{l}\mathrm{(}R\mathrm{)}{\mathrm{)}}^{\prime}\\ \text{\hspace{1em}}=\text{\hspace{0.17em}}{\mathrm{sinh}}_{l}\mathrm{(}R\mathrm{)},{\mathrm{cosh}}_{l}^{2}\mathrm{(}R\mathrm{)}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{\mathrm{sinh}}_{l}^{2}\mathrm{(}R\mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}l,\text{\hspace{0.17em}}\\ \mathrm{(}{\mathrm{tanh}}_{l}\mathrm{(}R\mathrm{)}{\mathrm{)}}^{\prime}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}l{\text{sech}}_{l}^{2}\mathrm{(}R\mathrm{)},\text{\hspace{0.17em}}\mathrm{(}{\text{sech}}_{l}\mathrm{(}R\mathrm{)}{\mathrm{)}}^{\prime}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{\mathrm{tanh}}_{l}\mathrm{(}R\mathrm{)}{\text{sech}}_{l}\mathrm{(}R\mathrm{)},\\ {\mathrm{tanh}}_{l}^{2}\mathrm{(}R\mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1\text{\hspace{0.17em}}-\text{\hspace{0.17em}}l{\text{sech}}_{l}^{2}\mathrm{(}R\mathrm{)},\text{\hspace{0.17em}}{\mathrm{coth}}_{l}^{2}\mathrm{(}R\mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1\text{\hspace{0.17em}}+\text{\hspace{0.17em}}l{\text{csch}}_{l}^{2}\mathrm{(}R\mathrm{)}.\end{array}\text{\hspace{1em}(18)}$$(18)

Correspondingly, we can define *l*-deformed triangular functions as follows:

$$\begin{array}{l}{\mathrm{sin}}_{l}\mathrm{(}R\mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{{e}^{iR}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}l{e}^{\text{\hspace{0.22em}}-iR}}{2i},\text{\hspace{0.17em}}{\mathrm{cos}}_{l}\mathrm{(}R\mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{{e}^{iR}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}l{e}^{-iR}}{2},\text{\hspace{0.17em}}\\ {\mathrm{tan}}_{l}\mathrm{(}R\mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{{\mathrm{sin}}_{l}\mathrm{(}R\mathrm{)}}{{\mathrm{cos}}_{l}\mathrm{(}R\mathrm{)}},\text{\hspace{0.17em}}{\mathrm{sec}}_{l}\mathrm{(}R\mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{1}{{\mathrm{cos}}_{l}\mathrm{(}R\mathrm{)}}.\end{array}\text{\hspace{1em}(19)}$$(19)

They satisfy the following formulas:

$$\begin{array}{l}\mathrm{(}{\mathrm{sin}}_{l}\mathrm{(}R\mathrm{)}{\mathrm{)}}^{\prime}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\mathrm{cos}}_{l}\mathrm{(}R\mathrm{)},\text{\hspace{0.17em}}\mathrm{(}{\mathrm{cos}}_{l}\mathrm{(}R\mathrm{)}{\mathrm{)}}^{\prime}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{\mathrm{sin}}_{l}\mathrm{(}R\mathrm{)},\text{\hspace{0.17em}}\mathrm{(}{\mathrm{tan}}_{l}\mathrm{(}R\mathrm{)}{\mathrm{)}}^{\prime}\\ \text{\hspace{1em}}=\text{\hspace{0.17em}}l{\text{sec}}_{l}^{2}\mathrm{(}R\mathrm{)},\\ \mathrm{(}{\mathrm{sec}}_{l}\mathrm{(}R\mathrm{)}{\mathrm{)}}^{\prime}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\mathrm{tan}}_{l}\mathrm{(}R\mathrm{)}{\mathrm{sec}}_{l}\mathrm{(}R\mathrm{)},\text{\hspace{0.17em}}{\mathrm{cos}}_{l}^{2}\mathrm{(}R\mathrm{)}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{\mathrm{sin}}_{l}^{2}\mathrm{(}R\mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}l,\\ 1\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{\mathrm{tan}}_{l}^{2}\mathrm{(}R\mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}l{\text{sec}}_{l}^{2}\mathrm{(}R\mathrm{)},\text{\hspace{0.17em}}1\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{\mathrm{cot}}_{l}^{2}\mathrm{(}R\mathrm{)}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}l{\text{csc}}_{l}^{2}\mathrm{(}R\mathrm{)}.\end{array}\text{\hspace{1em}(20)}$$(20)

**Case 3:** For *l*_{0}=0, solution (12) can be obtained.

**Family 3**

From the above result (7) and some typical solutions of (4) with *l*_{1}=0 in the study of Dai and Wang [22], we have

**Case 1:** For *l*_{0}*l*_{2}=−1,

$$\begin{array}{c}{u}_{10}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{3{p}_{x}{q}_{y}}{A}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{p}_{x}{q}_{y}}{2A}[{\mathrm{tanh}}_{{l}_{0}}^{2}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{\mathrm{tanh}}_{{l}_{0}}^{-2}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}],\\ {v}_{10}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{3{p}_{xx}^{2}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}8{p}_{x}^{4}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}4{p}_{x}{p}_{t}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}4{p}_{x}{p}_{xxx}}{8A{p}_{x}^{2}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\frac{3{p}_{xx}}{2A}[{\mathrm{tanh}}_{{l}_{0}}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}\\ +\text{\hspace{0.17em}}{\mathrm{tanh}}_{{l}_{0}}^{-1}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}]\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{p}_{x}^{2}}{2A}[{\mathrm{tanh}}_{{l}_{0}}^{2}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{\mathrm{tanh}}_{{l}_{0}}^{-2}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}],\end{array}\text{\hspace{1em}(21)}$$(21)

**Case 2:** For *l*_{0}*l*_{2}=1

$$\begin{array}{c}{u}_{11}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{p}_{x}{q}_{y}}{A}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{p}_{x}{q}_{y}}{2A}[{\mathrm{tan}}_{{l}_{0}}^{2}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{\mathrm{tan}}_{{l}_{0}}^{-2}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}],\\ {v}_{11}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{3{p}_{xx}^{2}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}8{p}_{x}^{4}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}4{p}_{x}{p}_{t}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}4{p}_{x}{p}_{xxx}}{8A{p}_{x}^{2}}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{p}_{xx}}{2A}[{\mathrm{tan}}_{{l}_{0}}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}\\ +\text{\hspace{0.17em}}{\mathrm{tan}}_{{l}_{0}}^{-1}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}]\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{3{p}_{x}^{2}}{2A}[{\mathrm{tan}}_{{l}_{0}}^{2}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}]\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{\mathrm{tan}}_{{l}_{0}}^{-2}\mathrm{(}p\text{\hspace{0.17em}}+\text{\hspace{0.17em}}q\mathrm{)}],\end{array}\text{\hspace{1em}(22)}$$(22)

where *p*≡ *p*(*x*, *t*) and *q*=*q*(*y*) are arbitrary functions of {*x*, *t*} and {*y*} respectively, *l*-deformed functions have the properties in (17)–(20).

**Case 3:** For *l*_{0}=0, we can derive solution (12).

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