In this section, we construct some new Jacobi elliptic exact solutions of some nonlinear partial fractional differential equations via the time-space fractional nonlinear KdV equation and the time-space fractional nonlinear Zakharov-Kunzetsov-Benjamin-Bona-Mahomy equation using the modified extended proposed algebraic method which has been paid attention to by many authors.

### 4.1 Example 1: Jacobi elliptic solutions for nonlinear fractional KdV equation

In this section, to demonstrate the effectiveness of this method, we use the complex transformation (9) to converting the nonlinear KdV equation with time-space fractional derivatives (1) to an ordinary differential equation; and we integrate twice, to find

\frac{1}{2}L{U}^{2}+\frac{a}{6}K{U}^{3}+\frac{{K}^{3}}{2}{\left[{U}^{\prime}\right]}^{2}+{C}_{1}U+{C}_{2}=0,

(13)

where {C}_{1} and {C}_{2} are the integration constants. Considering the homogeneous balance between the highest order derivatives and the nonlinear terms in (13), we get

U(\xi )={\alpha}_{0}+{\alpha}_{1}\varphi (\xi )+{\alpha}_{2}{\varphi}^{2}(\xi )+\frac{{\alpha}_{3}}{\varphi (\xi )}+\frac{{\alpha}_{4}}{{\varphi}^{2}(\xi )},

(14)

where {\alpha}_{0}, {\alpha}_{1}, {\alpha}_{2}, {a}_{3}, {a}_{4}, *L*, and *K* are arbitrary constants to be determined later. Substituting (14) and (12) into (13), collecting all terms of \varphi (\xi ), and then setting each coefficient \varphi (\xi ) to be zero, we get a system of algebraic equations. With the aid of Maple or Mathematica we can solve this system of algebraic equations to obtain the following cases of solutions:

Case 1.

\begin{array}{r}{a}_{0}=\frac{2{C}_{1}(L+4{K}^{3}{e}_{1})}{192{K}^{6}{e}_{0}{e}_{2}-{L}^{2}+16{K}^{6}{e}_{1}^{2}},\phantom{\rule{2em}{0ex}}{a}_{2}=\frac{24{C}_{1}{K}^{3}{e}_{2}}{192{K}^{6}{e}_{0}{e}_{2}-{L}^{2}+16{K}^{6}{e}_{1}^{2}},\\ a=-\frac{192{K}^{6}{e}_{0}{e}_{2}-{L}^{2}+16{K}^{6}{e}_{1}^{2}}{4K{C}_{1}},\phantom{\rule{2em}{0ex}}{a}_{4}=\frac{24{C}_{1}{K}^{3}{e}_{0}}{192{K}^{6}{e}_{0}{e}_{2}-{L}^{2}+16{K}^{6}{e}_{1}^{2}},\\ {C}_{2}=\frac{2{C}_{1}^{2}(-576L{e}_{2}{K}^{4}{e}_{0}+4\text{,}608{K}^{9}{e}_{0}{e}_{1}{e}_{2}+{L}^{3}-48{K}^{6}{e}_{1}^{2}L-128{K}^{9}{e}_{1}^{3})}{3{(192{K}^{6}{e}_{0}{e}_{2}-{L}^{2}+16{K}^{6}{e}_{1}^{2})}^{2}},\\ {a}_{1}={a}_{3}=0,\end{array}

(15)

where {C}_{1}, *L*, *K*, {e}_{0}, {e}_{1}, and {e}_{2} are arbitrary constants.

Case 2.

\begin{array}{c}{C}_{1}=\frac{1}{2}(192{K}^{6}{e}_{0}{e}_{2}-{L}^{2}+16{K}^{6}{e}_{1}^{2})\hfill \\ \phantom{{C}_{1}=}\times \sqrt{\frac{6{C}_{2}}{-576L{e}_{2}{K}^{6}{e}_{0}+4\text{,}608{K}^{9}{e}_{2}{e}_{1}{e}_{0}+{L}^{3}-48{K}^{6}{e}_{1}^{2}L-128{K}^{9}{e}_{1}^{3}}},\hfill \\ {a}_{0}=(L+4{K}^{3}{e}_{1})\sqrt{\frac{6{C}_{2}}{-576L{e}_{2}{K}^{6}{e}_{0}+4\text{,}608{K}^{9}{e}_{2}{e}_{1}{e}_{0}+{L}^{3}-48{K}^{6}{e}_{1}^{2}L-128{K}^{9}{e}_{1}^{3}}},\hfill \\ {a}_{2}=12{K}^{3}{e}_{2}\sqrt{\frac{6{C}_{2}}{-576L{e}_{2}{K}^{6}{e}_{0}+4\text{,}608{K}^{9}{e}_{2}{e}_{1}{e}_{0}+{L}^{3}-48{K}^{6}{e}_{1}^{2}L-128{K}^{9}{e}_{1}^{3}}},\hfill \\ a=-\frac{1}{12K\sqrt{{C}_{2}}}\sqrt{6[-576L{e}_{2}{K}^{6}{e}_{0}+4\text{,}608{K}^{9}{e}_{2}{e}_{1}{e}_{0}+{L}^{3}-48{K}^{6}{e}_{1}^{2}L-128{K}^{9}{e}_{1}^{3}]},\hfill \\ {a}_{4}=12{K}^{3}{e}_{0}\sqrt{\frac{6{C}_{2}}{-576L{e}_{2}{K}^{6}{e}_{0}+4\text{,}608{K}^{9}{e}_{2}{e}_{1}{e}_{0}+{L}^{3}-48{K}^{6}{e}_{1}^{2}L-128{K}^{9}{e}_{1}^{3}}},\hfill \\ {a}_{1}={a}_{3}=0,\hfill \end{array}

(16)

where {e}_{0}, {e}_{1}, {e}_{2}, *K*, *L*, and {C}_{2} are arbitrary constants.

Note that there are other cases which are omitted here for convenience. Since the solutions obtained here are so many, we just list some of the Jacobi exact solutions corresponding to Case 1 to illustrate the effectiveness of the proposed method. Substituting (15) into (14) we have

\begin{array}{rcl}u& =& \frac{2{C}_{1}(L+4{K}^{3}{e}_{1})}{192{K}^{6}{e}_{0}{e}_{2}-{L}^{2}+16{K}^{6}{e}_{1}^{2}}+\frac{24{C}_{1}{K}^{3}{e}_{2}}{192{K}^{6}{e}_{0}{e}_{2}-{L}^{2}+16{K}^{6}{e}_{1}^{2}}{\varphi}^{2}(\xi )\\ +\frac{24{C}_{1}{K}^{3}{e}_{0}}{[192{K}^{6}{e}_{0}{e}_{2}-{L}^{2}+16{K}^{6}{e}_{1}^{2}]{\varphi}^{2}(\xi )},\end{array}

(17)

where

\xi =\frac{K{x}^{\beta}}{\mathrm{\Gamma}(\beta +1)}+\frac{L{t}^{\alpha}}{\mathrm{\Gamma}(\alpha +1)}.

According to the general solutions of (12) which are discussed in Table 1, we have the following families of exact solutions:

Family 1. If {e}_{0}=1, {e}_{1}=-(1+{m}^{2}), {e}_{2}={m}^{2} the exact traveling wave solution takes the form

\begin{array}{rcl}{u}_{1}& =& \frac{2{C}_{1}[L-4{K}^{3}(1+{m}^{2})]}{192{K}^{6}{m}^{2}-{L}^{2}+16{K}^{6}{(1+{m}^{2})}^{2}}\\ +\frac{24{C}_{1}{K}^{3}{m}^{2}}{192{K}^{6}{m}^{2}-{L}^{2}+16{K}^{6}{(1+{m}^{2})}^{2}}s{n}^{2}[\frac{K{x}^{\beta}}{\mathrm{\Gamma}(\beta +1)}+\frac{L{t}^{\alpha}}{\mathrm{\Gamma}(\alpha +1)}]\\ +\frac{24{C}_{1}{K}^{3}}{[192{K}^{6}{m}^{2}-{L}^{2}+16{K}^{6}{(1+{m}^{2})}^{2}]}n{s}^{2}[\frac{K{x}^{\beta}}{\mathrm{\Gamma}(\beta +1)}+\frac{L{t}^{\alpha}}{\mathrm{\Gamma}(\alpha +1)}].\end{array}

(18)

To illustrate the behavior of the Jacobi elliptic solution {u}_{1} (18), see Figure 1.

Furthermore,

\begin{array}{rcl}{u}_{2}& =& \frac{2{C}_{1}[L-4{K}^{3}(1+{m}^{2})]}{192{K}^{6}{m}^{2}-{L}^{2}+16{K}^{6}{(1+{m}^{2})}^{2}}\\ +\frac{24{C}_{1}{K}^{3}{m}^{2}}{192{K}^{6}{m}^{2}-{L}^{2}+16{K}^{6}{(1+{m}^{2})}^{2}}c{d}^{2}[\frac{K{x}^{\beta}}{\mathrm{\Gamma}(\beta +1)}+\frac{L{t}^{\alpha}}{\mathrm{\Gamma}(\alpha +1)}]\\ +\frac{24{C}_{1}{K}^{3}}{[192{K}^{6}{m}^{2}-{L}^{2}+16{K}^{6}{(1+{m}^{2})}^{2}]}d{c}^{2}[\frac{K{x}^{\beta}}{\mathrm{\Gamma}(\beta +1)}+\frac{L{t}^{\alpha}}{\mathrm{\Gamma}(\alpha +1)}],\end{array}

(19)

where {C}_{2}=\frac{2{C}_{1}^{2}[-576L{m}^{2}{K}^{4}-4\text{,}608{K}^{9}{m}^{2}(1+{m}^{2})+{L}^{3}-48{K}^{6}{(1+{m}^{2})}^{2}L+128{K}^{9}{(1+{m}^{2})}^{3}]}{3{[192{K}^{6}{m}^{2}-{L}^{2}+16{K}^{6}{(1+{m}^{2})}^{2}]}^{2}}.

Family 2. If {e}_{0}=1-{m}^{2}, {e}_{1}=2{m}^{2}-1, {e}_{2}=-{m}^{2}, the exact traveling wave solution takes the form

\begin{array}{rcl}{u}_{3}& =& \frac{2{C}_{1}(L+4{K}^{3}(2{m}^{2}-1))}{\{-192{K}^{6}(1-{m}^{2}){m}^{2}-{L}^{2}+16{K}^{6}{(2{m}^{2}-1)}^{2}\}}\\ -\frac{24{C}_{1}{K}^{3}{m}^{2}c{n}^{2}[\frac{K{x}^{\beta}}{\mathrm{\Gamma}(\beta +1)}+\frac{L{t}^{\alpha}}{\mathrm{\Gamma}(\alpha +1)}]}{\{-192{K}^{6}(1-{m}^{2}){m}^{2}-{L}^{2}+16{K}^{6}{(2{m}^{2}-1)}^{2}\}}\\ +\frac{24{C}_{1}{K}^{3}(1-{m}^{2})n{c}^{2}[\frac{K{x}^{\beta}}{\mathrm{\Gamma}(\beta +1)}+\frac{L{t}^{\alpha}}{\mathrm{\Gamma}(\alpha +1)}]}{\{-192{K}^{6}(1-{m}^{2}){m}^{2}-{L}^{2}+16{K}^{6}{(2{m}^{2}-1)}^{2}\}},\end{array}

(20)

where {C}_{2} = \frac{1}{3{[-192{K}^{6}{m}^{2}(1-{m}^{2})-{L}^{2}+16{K}^{6}{(2{m}^{2}-1)}^{2}]}^{2}}{2{C}_{1}^{2}(576L{m}^{2}{K}^{4}(2{m}^{2}-1) − 4\text{,}608{K}^{9}{m}^{2}(2{m}^{2}-1)(1-{m}^{2}) + {L}^{3} − 48{K}^{6}{(2{m}^{2}-1)}^{2}L − 128{K}^{9}{(2{m}^{2}-1)}^{3})}.

To illustrate the behavior of the Jacobi elliptic solution {u}_{3} (20), see Figure 2.

Family 3. If {e}_{0}={m}^{2}-1, {e}_{1}=2-{m}^{2}, {e}_{2}=-1, the exact traveling wave solution takes the form

\begin{array}{rcl}{u}_{4}& =& \frac{2{C}_{1}[L+4{K}^{3}(2-{m}^{2})]}{[192{K}^{6}(1-{m}^{2})-{L}^{2}+16{K}^{6}{(2-{m}^{2})}^{2}]}\\ -\frac{24{C}_{1}{K}^{3}d{n}^{2}[\frac{K{x}^{\beta}}{\mathrm{\Gamma}(\beta +1)}+\frac{L{t}^{\alpha}}{\mathrm{\Gamma}(\alpha +1)}]}{[192{K}^{6}(1-{m}^{2})-{L}^{2}+16{K}^{6}{(2-{m}^{2})}^{2}]}\\ +\frac{24{C}_{1}{K}^{3}({m}^{2}-1)n{d}^{2}[\frac{K{x}^{\beta}}{\mathrm{\Gamma}(\beta +1)}+\frac{L{t}^{\alpha}}{\mathrm{\Gamma}(\alpha +1)}]}{[192{K}^{6}(1-{m}^{2})-{L}^{2}+16{K}^{6}{(2-{m}^{2})}^{2}]},\end{array}

(21)

where {C}_{2}=\frac{2{C}_{1}^{2}(576L{K}^{4}({m}^{2}-1)-4\text{,}608{K}^{9}({m}^{2}-1)(2-{m}^{2})+{L}^{3}-48{K}^{6}{(2-{m}^{2})}^{2}L-128{K}^{9}{(2-{m}^{2})}^{3})}{3{[192{K}^{6}(1-{m}^{2})-{L}^{2}+16{K}^{6}{(2-{m}^{2})}^{2}]}^{2}}.

Family 4. If {e}_{0}=\frac{1}{4}(1-{m}^{2}), {e}_{1}=\frac{1}{2}(1+{m}^{2}), {e}_{2}=\frac{1}{4}(1-{m}^{2}), the exact traveling wave solution takes the form

\begin{array}{rcl}{u}_{5}& =& \frac{2{C}_{1}(L+2{K}^{3}(1+{m}^{2}))}{[12{K}^{6}{(1-{m}^{2})}^{2}-{L}^{2}+4{K}^{6}{(1+{m}^{2})}^{2}]}\\ +\frac{6{C}_{1}{K}^{3}(1-{m}^{2}){[nc(\frac{K{x}^{\beta}}{\mathrm{\Gamma}(\beta +1)}+\frac{L{t}^{\alpha}}{\mathrm{\Gamma}(\alpha +1)})\pm sc(\frac{K{x}^{\beta}}{\mathrm{\Gamma}(\beta +1)}+\frac{L{t}^{\alpha}}{\mathrm{\Gamma}(\alpha +1)})]}^{2}}{[12{K}^{6}{(1-{m}^{2})}^{2}-{L}^{2}+4{K}^{6}{(1+{m}^{2})}^{2}]}\\ +\frac{6{C}_{1}{K}^{3}(1-{m}^{2})}{[12{K}^{6}{(1-{m}^{2})}^{2}-{L}^{2}+4{K}^{6}{(1+{m}^{2})}^{2}]}\\ \times \frac{1}{{[nc(\frac{K{x}^{\beta}}{\mathrm{\Gamma}(\beta +1)}+\frac{L{t}^{\alpha}}{\mathrm{\Gamma}(\alpha +1)})\pm sc(\frac{K{x}^{\beta}}{\mathrm{\Gamma}(\beta +1)}+\frac{L{t}^{\alpha}}{\mathrm{\Gamma}(\alpha +1)})]}^{2}},\end{array}

(22)

where {C}_{2}=\frac{2{C}_{1}^{2}(-36L{K}^{4}{(1-{m}^{2})}^{2}+144{K}^{9}{(1-{m}^{2})}^{2}(1+{m}^{2})+{L}^{3}-12{K}^{6}{(1+{m}^{2})}^{2}L-16{K}^{9}{(1+{m}^{2})}^{3})}{3{[12{K}^{6}{(1-{m}^{2})}^{2}-{L}^{2}+4{K}^{6}{(1+{m}^{2})}^{2}]}^{2}}.

To illustrate the behavior of the Jacobi elliptic solution {u}_{5} (22), see Figure 3.

Family 5. If {e}_{0}=\frac{{m}^{2}}{4}, {e}_{1}=\frac{1}{2}({m}^{2}-2), {e}_{2}=\frac{{m}^{2}}{4}, the exact traveling wave solution takes the form

\begin{array}{rcl}{u}_{6}& =& \frac{2{C}_{1}(L+2{K}^{3}({m}^{2}-2))}{12{K}^{6}{m}^{4}-{L}^{2}+4{K}^{6}{({m}^{2}-2)}^{2}}\\ +\frac{6{C}_{1}{K}^{3}{m}^{2}{[sn(\frac{K{x}^{\beta}}{\mathrm{\Gamma}(\beta +1)}+\frac{L{t}^{\alpha}}{\mathrm{\Gamma}(\alpha +1)})\pm icn(\frac{K{x}^{\beta}}{\mathrm{\Gamma}(\beta +1)}+\frac{L{t}^{\alpha}}{\mathrm{\Gamma}(\alpha +1)})]}^{2}}{[12{K}^{6}{m}^{4}-{L}^{2}+4{K}^{6}{({m}^{2}-2)}^{2}]}\\ +\frac{24{C}_{1}{K}^{3}{m}^{2}}{[12{K}^{6}{m}^{4}-{L}^{2}+4{K}^{6}{({m}^{2}-2)}^{2}]}\\ \times \frac{1}{{[sn(\frac{K{x}^{\beta}}{\mathrm{\Gamma}(\beta +1)}+\frac{L{t}^{\alpha}}{\mathrm{\Gamma}(\alpha +1)})\pm icn(\frac{K{x}^{\beta}}{\mathrm{\Gamma}(\beta +1)}+\frac{L{t}^{\alpha}}{\mathrm{\Gamma}(\alpha +1)})]}^{2}},\end{array}

(23)

where {C}_{2}=\frac{2{C}_{1}^{2}(-36L{K}^{4}{m}^{4}+144{K}^{9}{m}^{4}({m}^{2}-2)+{L}^{3}-12{K}^{6}{({m}^{2}-2)}^{2}L-16{K}^{9}{({m}^{2}-2)}^{3})}{3{[12{K}^{6}{m}^{4}-{L}^{2}+4{K}^{6}{({m}^{2}-2)}^{2}]}^{2}}.

To illustrate the behavior of the Jacobi elliptic solution {u}_{6} (23), see Figures 4 and 5.

Similarly, we can write down the other families of exact solutions of (1) which are omitted for convenience.

### 4.2 Example 2: Jacobi elliptic solutions for nonlinear fractional ZKBBM equation

In this section we use the proposed method to find the Jacobi elliptic solutions for the nonlinear fractional ZKBBM equation with time and space fractional derivatives (2). The complex fractional transformations (9) convert the nonlinear fractional ZKBBM equation (2) to the following nonlinear ordinary differential equation:

(L+K)U-aK{U}^{2}-b{K}^{2}{U}^{\u2033}+{C}_{1}=0,

(24)

where {C}_{1} is the integration constant. Considering the homogeneous balance between the highest order derivative and the nonlinear term in (24), we have

U(\xi )={\alpha}_{0}+{\alpha}_{1}\varphi (\xi )+{\alpha}_{2}{\varphi}^{2}(\xi )+\frac{{\alpha}_{3}}{\varphi (\xi )}+\frac{{\alpha}_{4}}{{\varphi}^{2}(\xi )},

(25)

where {\alpha}_{0}, {\alpha}_{1}, {\alpha}_{2}, {a}_{3}, {a}_{4}, *L*, and *K* are arbitrary constants to be determined later. Substituting (25) and (12) into (24), collecting all the terms of powers of \varphi (\xi ) and setting each coefficient \varphi (\xi ) to zero, we get a system of algebraic equations. With the aid of Maple or Mathematica we can solve this system of algebraic equations to obtain the following sets of solutions:

Case 1.

\begin{array}{c}{a}_{0}=-\frac{4b{K}^{2}L{e}_{1}-K-L}{2aK},\phantom{\rule{2em}{0ex}}{a}_{2}=-\frac{6KbL{e}_{2}}{a},\phantom{\rule{2em}{0ex}}{a}_{4}=-\frac{6KbL{e}_{0}}{a},\hfill \\ {C}_{1}=\frac{1}{4aK}\{-{K}^{2}-2LK+192{b}^{2}{K}^{4}{L}^{2}{e}_{0}{e}_{2}-{L}^{2}+16{b}^{2}{K}^{4}{L}^{2}{e}_{1}^{2}\},\hfill \\ {a}_{1}={a}_{3}=0,\hfill \end{array}

(26)

where *b*, *L*, *K*, {e}_{0}, {e}_{1}, and {e}_{2} are arbitrary nonzero constants. There are many other cases which are omitted for convenience.

Substituting (26) into (25) we have

u=-\frac{4b{K}^{2}L{e}_{1}-K-L}{2aK}-\frac{6KbL{e}_{2}}{a}{\varphi}^{2}(\xi )-\frac{6KbL{e}_{0}}{a{\varphi}^{2}(\xi )},

(27)

where

\xi =\frac{K{x}^{\beta}}{\mathrm{\Gamma}(\beta +1)}+\frac{L{t}^{\alpha}}{\mathrm{\Gamma}(\alpha +1)}

(28)

and

{C}_{1}=\frac{1}{4aK}\{-{K}^{2}-2LK+192{b}^{2}{K}^{4}{L}^{2}{e}_{0}{e}_{2}-{L}^{2}+16{b}^{2}{K}^{4}{L}^{2}{e}_{1}^{2}\}.

According to the general solutions of (12) which are discussed in Table 1, we have the following families of Jacobi elliptic exact solutions to the nonlinear ZKBBM equation:

Family 1. If {e}_{0}=1, {e}_{1}=-(1+{m}^{2}), {e}_{2}={m}^{2}, the exact traveling wave solution takes the form

\begin{array}{rcl}{u}_{1}& =& \frac{4b{K}^{2}L(1+{m}^{2})+K+L}{2aK}-\frac{6KbL{m}^{2}}{a}s{n}^{2}[\frac{K{x}^{\beta}}{\mathrm{\Gamma}(\beta +1)}+\frac{L{t}^{\alpha}}{\mathrm{\Gamma}(\alpha +1)}]\\ -\frac{6KbL}{a}n{s}^{2}[\frac{K{x}^{\beta}}{\mathrm{\Gamma}(\beta +1)}+\frac{L{t}^{\alpha}}{\mathrm{\Gamma}(\alpha +1)}].\end{array}

(29)

To illustrate the behavior of the Jacobi elliptic solution {u}_{1} (29), see Figure 6.

Furthermore

\begin{array}{rcl}{u}_{2}& =& \frac{4b{K}^{2}L(1+{m}^{2})+K+L}{2aK}-\frac{6KbL{m}^{2}}{a}c{d}^{2}[\frac{K{x}^{\beta}}{\mathrm{\Gamma}(\beta +1)}+\frac{L{t}^{\alpha}}{\mathrm{\Gamma}(\alpha +1)}]\\ -\frac{6KbL}{a}d{c}^{2}[\frac{K{x}^{\beta}}{\mathrm{\Gamma}(\beta +1)}+\frac{L{t}^{\alpha}}{\mathrm{\Gamma}(\alpha +1)}],\end{array}

(30)

where {C}_{1}=\frac{1}{4aK}\{-{K}^{2}-2LK+192{b}^{2}{K}^{4}{L}^{2}{m}^{2}-{L}^{2}+16{b}^{2}{K}^{4}{L}^{2}{(1+{m}^{2})}^{2}\}.

Family 2. If {e}_{0}=1-{m}^{2}, {e}_{1}=2{m}^{2}-1, {e}_{2}=-{m}^{2}, the exact traveling wave solution takes the form

\begin{array}{rcl}{u}_{3}& =& -\frac{4b{K}^{2}L(2{m}^{2}-1)-K-L}{2aK}+\frac{6KbL{m}^{2}}{a}c{n}^{2}(\frac{K{x}^{\beta}}{\mathrm{\Gamma}(\beta +1)}+\frac{L{t}^{\alpha}}{\mathrm{\Gamma}(\alpha +1)})\\ -\frac{6KbL(1-{m}^{2})}{a}n{c}^{2}(\frac{K{x}^{\beta}}{\mathrm{\Gamma}(\beta +1)}+\frac{L{t}^{\alpha}}{\mathrm{\Gamma}(\alpha +1)}),\end{array}

(31)

where {C}_{1}=\frac{1}{4aK}\{-{K}^{2}-2LK-192{b}^{2}{K}^{4}{L}^{2}(1-{m}^{2}){m}^{2}-{L}^{2}+16{b}^{2}{K}^{4}{L}^{2}{(2{m}^{2}-1)}^{2}\}.

To illustrate the behavior of the Jacobi elliptic solution {u}_{3} (31), see Figure 7.

Family 3. If {e}_{0}={m}^{2}-1, {e}_{1}=2-{m}^{2}, {e}_{2}=-1, the exact traveling wave solution takes the form

\begin{array}{rcl}{u}_{4}& =& -\frac{4b{K}^{2}L(2-{m}^{2})-K-L}{2aK}+\frac{6KbL}{a}d{n}^{2}[\frac{K{x}^{\beta}}{\mathrm{\Gamma}(\beta +1)}+\frac{L{t}^{\alpha}}{\mathrm{\Gamma}(\alpha +1)}]\\ -\frac{6KbL({m}^{2}-1)}{a}n{d}^{2}[\frac{K{x}^{\beta}}{\mathrm{\Gamma}(\beta +1)}+\frac{L{t}^{\alpha}}{\mathrm{\Gamma}(\alpha +1)}],\end{array}

(32)

where {C}_{1}=\frac{1}{4aK}\{-{K}^{2}-2LK-192{b}^{2}{K}^{4}{L}^{2}({m}^{2}-1)-{L}^{2}+16{b}^{2}{K}^{4}{L}^{2}{(2-{m}^{2})}^{2}\}.

Family 4. If {e}_{0}=\frac{1}{4}(1-{m}^{2}), {e}_{1}=\frac{1}{2}(1+{m}^{2}), {e}_{2}=\frac{1}{4}(1-{m}^{2}), the exact traveling wave solution takes the form

\begin{array}{rcl}{u}_{5}& =& -\frac{2b{K}^{2}L(1+{m}^{2})-K-L}{2aK}\\ -\frac{3KbL(1-{m}^{2})}{2a}{[nc(\frac{K{x}^{\beta}}{\mathrm{\Gamma}(\beta +1)}+\frac{L{t}^{\alpha}}{\mathrm{\Gamma}(\alpha +1)})\pm sc(\frac{K{x}^{\beta}}{\mathrm{\Gamma}(\beta +1)}+\frac{L{t}^{\alpha}}{\mathrm{\Gamma}(\alpha +1)})]}^{2}\\ -\frac{3KbL(1-{m}^{2})}{2a{[nc(\frac{K{x}^{\beta}}{\mathrm{\Gamma}(\beta +1)}+\frac{L{t}^{\alpha}}{\mathrm{\Gamma}(\alpha +1)})\pm sc(\frac{K{x}^{\beta}}{\mathrm{\Gamma}(\beta +1)}+\frac{L{t}^{\alpha}}{\mathrm{\Gamma}(\alpha +1)})]}^{2}},\end{array}

(33)

where {C}_{1}=\frac{1}{4aK}\{-{K}^{2}-2LK+12{b}^{2}{K}^{4}{L}^{2}{(1-{m}^{2})}^{2}-{L}^{2}+4{b}^{2}{K}^{4}{L}^{2}{(1+{m}^{2})}^{2}\}.

To illustrate the behavior of the Jacobi elliptic solution {u}_{5} (33), see Figure 8.

Family 5. If {e}_{0}=\frac{{m}^{2}}{4}, {e}_{1}=\frac{1}{2}({m}^{2}-2), {e}_{2}=\frac{{m}^{2}}{4}, the exact traveling wave solution takes the form

\begin{array}{rl}{u}_{6}=& -\frac{2b{K}^{2}L({m}^{2}-2)-K-L}{2aK}\\ -\frac{3KbL{m}^{2}}{2a}{[sn(\frac{K{x}^{\beta}}{\mathrm{\Gamma}(\beta +1)}+\frac{L{t}^{\alpha}}{\mathrm{\Gamma}(\alpha +1)})\pm icn(\frac{K{x}^{\beta}}{\mathrm{\Gamma}(\beta +1)}+\frac{L{t}^{\alpha}}{\mathrm{\Gamma}(\alpha +1)})]}^{2}\\ -\frac{3KbL{m}^{2}}{2a{[sn(\frac{K{x}^{\beta}}{\mathrm{\Gamma}(\beta +1)}+\frac{L{t}^{\alpha}}{\mathrm{\Gamma}(\alpha +1)})\pm icn(\frac{K{x}^{\beta}}{\mathrm{\Gamma}(\beta +1)}+\frac{L{t}^{\alpha}}{\mathrm{\Gamma}(\alpha +1)})]}^{2}},\end{array}

(34)

where {C}_{1}=\frac{1}{4aK}\{-{K}^{2}-2LK+12{b}^{2}{K}^{4}{L}^{2}{m}^{4}-{L}^{2}+4{b}^{2}{K}^{4}{L}^{2}{({m}^{2}-2)}^{2}\}.

To illustrate the behavior of the Jacobi elliptic solution {u}_{6} (34), see Figures 9 and 10.

Similarly, we can write down the other families of exact solutions of (2) which are omitted for convenience.