Nonlinear wave solutions of the Kudryashov–Sinelshchikov dynamical equation in mixtures liquid-gas bubbles under the consideration of heat transfer and viscosity

Aly R. SeadawyMathematics Department, Faculty of Science, Taibah University, Al-Madinah Al-Munawarah, Saudi ArabiaCorrespondence[email protected]

https://orcid.org/0000-0002-7412-4773

Mujahid IqbalFaculty of Science, Jiangsu University, Zhenjiang, People's Republic of China

https://orcid.org/0000-0003-1234-4321

Dianchen LuFaculty of Science, Jiangsu University, Zhenjiang, People's Republic of China

Figures & data

Figure 1. (a) Dark soliton solution for Equation (Equation15(15) $\begin{aligned} u_{1} (x, t) & = β_{1} d_{2} (ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1) + \frac{β_{1} d_{2} ϵ^{2} c s c h {[\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})]}^{4}}{4 {(ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}} \\ - \frac{b_{1} β_{1} (ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}{β_{2}} - \frac{β_{1}^{2} β_{3} d_{2} {(ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}}{β_{2}^{2}} \\ - \frac{\sqrt{b_{1}^{2} β_{2}^{2} κ^{2} + 4 β_{1} (β_{1} - b_{1}^{2} β_{3} κ^{2})} - b_{1} β_{2} κ + 2 β_{1} (β_{1} d_{2} κ + 1)}{2 β_{1} κ} . \end{aligned}$ (15) ) and (b) bright soliton solution for Equation (Equation16(16) $\begin{aligned} u_{2} (x, t) & = \frac{1}{4} (\frac{4 β_{1} d_{2} {(η ϵ \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ)}^{2}}{{(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} {(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}} \\ - 2 b_{1} \sqrt{\frac{β_{1}}{β_{3}}} (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1) - β_{1} d_{2} {(\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}^{2} \\ + 2 β_{2} \sqrt{\frac{β_{1}}{β_{3}}} d_{2} (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1) \\ - \frac{2 (\sqrt{β_{2}^{2} b_{1}^{2} κ^{2} - 4 β_{1} β_{3} b_{1}^{2} κ^{2} + 4 β_{1}^{2}} - b_{1} β_{2} κ + 2 β_{1} (β_{1} d_{2} κ + 1))}{β_{1} κ}) . \end{aligned}$ (16) ).

Figure 1. (a) Dark soliton solution for Equation (Equation15(15) u1(x,t)=β1d2ϵcoth12β1x−λt+ζ0+1+β1d2ϵ2csch12β1x−λt+ζ044ϵcoth12β1x−λt+ζ0+12−b1β1ϵcoth12β1x−λt+ζ0+1β2−β12β3d2ϵcoth12β1x−λt+ζ0+12β22−b12β22κ2+4β1β1−b12β3κ2−b1β2κ+2β1β1d2κ+12β1κ.(15) ) and (b) bright soliton solution for Equation (Equation16(16) u2(x,t)=144β1d2ηϵcoshβ1x−λt+ζ0+ϵ2η+coshβ1x−λt+ζ02η+coshβ1x−λt+ζ0+ϵsinhβ1x−λt+ζ02−2b1β1β3ϵsinhβ1x−λt+ζ0η+coshβ1x−λt+ζ0+1−β1d2ϵsinhβ1x−λt+ζ0η+coshβ1x−λt+ζ0+12+2β2β1β3d2ϵsinhβ1x−λt+ζ0η+coshβ1x−λt+ζ0+1−2β22b12κ2−4β1β3b12κ2+4β12−b1β2κ+2β1β1d2κ+1β1κ.(16) ).

Figure 2. (a) Bright soliton solution for Equation (Equation17(17) $\begin{aligned} u_{3} (x, t) & = b_{1} (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1) + (ϵ^{2} (η \sqrt{p^{2} + 1} \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + 1 \\ {- p \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} β_{1} d_{2})/ ({(η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} \\ {(η \sqrt{p^{2} + 1} + p ϵ + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}) \\ - β_{2} d_{2} (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1) \\ - β_{3} d_{2} {(\frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}^{2} \\ - \frac{\sqrt{b_{1}^{2} β_{2}^{2} κ^{2} + 4 β_{1} (β_{1} - b_{1}^{2} β_{3} κ^{2})} - b_{1} β_{2} κ + 2 β_{1} (β_{1} d_{2} κ + 1)}{2 β_{1} κ} . \end{aligned}$ (17) ), and (b) dark soliton solution for Equation (Equation19(19) $\begin{aligned} u_{4} (x, t) & = β_{1} d_{2} (ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1) + \frac{β_{1} d_{2} ϵ^{2} c s c h {[\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})]}^{4}}{4 {(ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}} \\ - \frac{b_{1} β_{1} (ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}{β_{2}} - \frac{β_{1}^{2} β_{3} d_{2} {(ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}}{β_{2}^{2}} \\ + \frac{\sqrt{b_{1}^{2} β_{2}^{2} κ^{2} + 4 β_{1} (β_{1} - b_{1}^{2} β_{3} κ^{2})} + b_{1} β_{2} κ - 2 β_{1} (β_{1} d_{2} κ + 1)}{2 β_{1} κ} . \end{aligned}$ (19) ).

Figure 2. (a) Bright soliton solution for Equation (Equation17(17) u3(x,t)=b1−ϵp+sinhβ1x−λt+ζ0ηp2+1+coshβ1x−λt+ζ0−1+ϵ2ηp2+1coshβ1x−λt+ζ0+1−psinhβ1x−λt+ζ02β1d2ηp2+1+coshβ1x−λt+ζ02ηp2+1+pϵ+coshβ1x−λt+ζ0+ϵsinhβ1x−λt+ζ02−β2d2−ϵp+sinhβ1x−λt+ζ0ηp2+1+coshβ1x−λt+ζ0−1−β3d2ϵp+sinhβ1x−λt+ζ0ηp2+1+coshβ1x−λt+ζ0+12−b12β22κ2+4β1β1−b12β3κ2−b1β2κ+2β1β1d2κ+12β1κ.(17) ), and (b) dark soliton solution for Equation (Equation19(19) u4(x,t)=β1d2ϵcoth12β1x−λt+ζ0+1+β1d2ϵ2csch12β1x−λt+ζ044ϵcoth12β1x−λt+ζ0+12−b1β1ϵcoth12β1x−λt+ζ0+1β2−β12β3d2ϵcoth12β1x−λt+ζ0+12β22+b12β22κ2+4β1β1−b12β3κ2+b1β2κ−2β1β1d2κ+12β1κ.(19) ).

Figure 3. (a) Combined dark–bright soliton solutions for Equation (Equation20(20) $\begin{aligned} u_{5} (x, t) & = \frac{1}{4} (\frac{4 β_{1} d_{2} {(η ϵ \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ)}^{2}}{{(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} {(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}} \\ β_{1} d_{2} (- {(\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}^{2}) - 2 b_{1} \sqrt{\frac{β_{1}}{β_{3}}} (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1) \\ + 2 β_{2} \sqrt{\frac{β_{1}}{β_{3}}} d_{2} (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1) \\ + \frac{2 (\sqrt{β_{2}^{2} b_{1}^{2} κ^{2} - 4 β_{1} β_{3} b_{1}^{2} κ^{2} + 4 β_{1}^{2}} + b_{1} β_{2} κ - 2 β_{1} (β_{1} d_{2} κ + 1))}{β_{1} κ}) . \end{aligned}$ (20) ) and (b) bright soliton solutions for Equation (Equation21(21) $\begin{aligned} u_{6} (x, t) & = b_{1} (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1) + (ϵ^{2} (η \sqrt{p^{2} + 1} \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + 1 \\ {- p \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} β_{1} d_{2})/ ({(η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} \\ {(η \sqrt{p^{2} + 1} + p ϵ + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}) \\ - β_{2} d_{2} (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1) \\ - β_{3} d_{2} {(\frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}^{2} \\ + \frac{\sqrt{b_{1}^{2} β_{2}^{2} κ^{2} + 4 β_{1} (β_{1} - b_{1}^{2} β_{3} κ^{2})} + b_{1} β_{2} κ - 2 β_{1} (β_{1} d_{2} κ + 1)}{2 β_{1} κ} . \end{aligned}$ (21) ).

Figure 3. (a) Combined dark–bright soliton solutions for Equation (Equation20(20) u5x,t=144β1d2ηϵcoshβ1x−λt+ζ0+ϵ2η+coshβ1x−λt+ζ02η+coshβ1x−λt+ζ0+ϵsinhβ1x−λt+ζ02β1d2−ϵsinhβ1x−λt+ζ0η+coshβ1x−λt+ζ0+12−2b1β1β3ϵsinhβ1x−λt+ζ0η+coshβ1x−λt+ζ0+1+2β2β1β3d2ϵsinhβ1x−λt+ζ0η+coshβ1x−λt+ζ0+1+2β22b12κ2−4β1β3b12κ2+4β12+b1β2κ−2β1β1d2κ+1β1κ.(20) ) and (b) bright soliton solutions for Equation (Equation21(21) u6(x,t)=b1−ϵp+sinhβ1x−λt+ζ0ηp2+1+coshβ1x−λt+ζ0−1+ϵ2ηp2+1coshβ1x−λt+ζ0+1−psinhβ1x−λt+ζ02β1d2ηp2+1+coshβ1x−λt+ζ02ηp2+1+pϵ+coshβ1x−λt+ζ0+ϵsinhβ1x−λt+ζ02−β2d2−ϵp+sinhβ1x−λt+ζ0ηp2+1+coshβ1x−λt+ζ0−1−β3d2ϵp+sinhβ1x−λt+ζ0ηp2+1+coshβ1x−λt+ζ0+12+b12β22κ2+4β1β1−b12β3κ2+b1β2κ−2β1β1d2κ+12β1κ.(21) ).

Figure 4. (a) Dark soliton solution for Equation (Equation23(23) $\begin{aligned} u_{7} (x, t) & = \frac{β_{1} d_{2} ϵ^{2} c s c h {[\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})]}^{4}}{4 {(ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}} + β_{1} d_{2} (ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1) \\ - \frac{2 β_{1} (A - 2 β_{1} β_{2} κ (ε + κ)) (ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}{β_{2} κ (β_{2}^{2} (ε^{2} - 4 κ^{2}) - 4 β_{1} β_{3} ε^{2})} \\ - \frac{β_{3} β_{1}^{2} d_{2} {(ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}}{β_{2}^{2}} \\ + \frac{- A β_{2} (ε - 2 κ) - 4 β_{3} β_{1}^{2} ε^{2} κ (β_{1} d_{2} ε - 1) + β_{2}^{2} β_{1} ε κ (ε - 2 κ) (β_{1} d_{2} (ε + 2 κ) + 1)}{β_{1} ε κ (4 β_{1} β_{3} ε^{2} - β_{2}^{2} (ε^{2} - 4 κ^{2}))} . \end{aligned}$ (23) ) and (b) Kink wave soliton solution for Equation (Equation24(24) $\begin{aligned} u_{8} (x, t) & = \frac{β_{1} d_{2} {(η ϵ \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ)}^{2}}{{(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} {(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}} \\ - \frac{1}{4} β_{1} d_{2} {(\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}^{2} + \frac{1}{2} β_{2} \sqrt{\frac{β_{1}}{β_{3}}} d_{2} (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1) \\ - \frac{\sqrt{\frac{β_{1}}{β_{3}}} (A - 2 β_{1} β_{2} κ (ε + κ)) (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}{κ (β_{2}^{2} (ε^{2} - 4 κ^{2}) - 4 β_{1} β_{3} ε^{2})} \\ + \frac{- A β_{2} (ε - 2 κ) - 4 β_{3} β_{1}^{2} ε^{2} κ (β_{1} d_{2} ε - 1) + β_{2}^{2} β_{1} ε κ (ε - 2 κ) (β_{1} d_{2} (ε + 2 κ) + 1)}{β_{1} ε κ (4 β_{1} β_{3} ε^{2} - β_{2}^{2} (ε^{2} - 4 κ^{2}))} . \end{aligned}$ (24) ).

Figure 4. (a) Dark soliton solution for Equation (Equation23(23) u7(x,t)=β1d2ϵ2csch12β1x−λt+ζ044ϵcoth12β1x−λt+ζ0+12+β1d2ϵcoth12β1x−λt+ζ0+1−2β1A−2β1β2κ(ε+κ)ϵcoth12β1x−λt+ζ0+1β2κβ22ε2−4κ2−4β1β3ε2−β3β12d2ϵcoth12β1x−λt+ζ0+12β22+−Aβ2(ε−2κ)−4β3β12ε2κβ1d2ε−1+β22β1εκ(ε−2κ)β1d2(ε+2κ)+1β1εκ4β1β3ε2−β22ε2−4κ2.(23) ) and (b) Kink wave soliton solution for Equation (Equation24(24) u8(x,t)=β1d2ηϵcoshβ1x−λt+ζ0+ϵ2η+coshβ1x−λt+ζ02η+coshβ1x−λt+ζ0+ϵsinhβ1x−λt+ζ02−14β1d2ϵsinhβ1x−λt+ζ0η+coshβ1x−λt+ζ0+12+12β2β1β3d2ϵsinhβ1x−λt+ζ0η+coshβ1x−λt+ζ0+1−β1β3A−2β1β2κ(ε+κ)ϵsinhβ1x−λt+ζ0η+coshβ1x−λt+ζ0+1κβ22ε2−4κ2−4β1β3ε2+−Aβ2(ε−2κ)−4β3β12ε2κβ1d2ε−1+β22β1εκ(ε−2κ)β1d2(ε+2κ)+1β1εκ4β1β3ε2−β22ε2−4κ2.(24) ).

Figure 5. (a,b) Solitary wave solutions for Equation (Equation25(25) $\begin{aligned} u_{9} (x, t) & = β_{1} d_{2} ϵ^{2} {(η \sqrt{p^{2} + 1} \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] - p \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}/ \\ ({(η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} \\ {(η \sqrt{p^{2} + 1} + p ϵ + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}) \\ - β_{2} d_{2} (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1) \\ - β_{3} d_{2} {(\frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}^{2} \\ + \frac{2 (A - 2 β_{1} β_{2} κ (ε + κ)) (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1)}{κ (β_{2}^{2} (ε^{2} - 4 κ^{2}) - 4 β_{1} β_{3} ε^{2})} \\ + \frac{- A β_{2} (ε - 2 κ) - 4 β_{3} β_{1}^{2} ε^{2} κ (β_{1} d_{2} ε - 1) + β_{2}^{2} β_{1} ε κ (ε - 2 κ) (β_{1} d_{2} (ε + 2 κ) + 1)}{β_{1} ε κ (β_{2}^{2} (4 κ^{2} - ε^{2}) + 4 β_{1} β_{3} ε^{2})} . \end{aligned}$ (25) ) and Equation (Equation27(27) $\begin{aligned} u_{10} (x, t) & = \frac{1}{4} β_{1} d_{2} ϵ (4 \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + \frac{ϵ c s c h {[\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})]}^{4}}{{(ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}}) \\ + \frac{β_{2} (\sqrt{b_{1}^{2} β_{2}^{2} κ^{2} + 4 β_{1}^{2}} - 2 β_{1}) + b_{1} κ (β_{2}^{2} - 2 β_{1}^{2} (ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1))}{2 β_{1} β_{2} κ} . \end{aligned}$ (27) ).

Figure 5. (a,b) Solitary wave solutions for Equation (Equation25(25) u9(x,t)=β1d2ϵ2ηp2+1coshβ1x−λt+ζ0−psinhβ1x−λt+ζ0+12ηp2+1+coshβ1x−λt+ζ02ηp2+1+pϵ+coshβ1x−λt+ζ0+ϵsinhβ1x−λt+ζ02−β2d2−ϵp+sinhβ1x−λt+ζ0ηp2+1+coshβ1x−λt+ζ0−1−β3d2ϵp+sinhβ1x−λt+ζ0ηp2+1+coshβ1x−λt+ζ0+12+2A−2β1β2κ(ε+κ)−ϵp+sinhβ1x−λt+ζ0ηp2+1+coshβ1x−λt+ζ0−1κβ22ε2−4κ2−4β1β3ε2+−Aβ2(ε−2κ)−4β3β12ε2κβ1d2ε−1+β22β1εκ(ε−2κ)β1d2(ε+2κ)+1β1εκβ224κ2−ε2+4β1β3ε2.(25) ) and Equation (Equation27(27) u10(x,t)=14β1d2ϵ4coth12β1x−λt+ζ0+ϵcsch12β1x−λt+ζ04ϵcoth12β1x−λt+ζ0+12+β2b12β22κ2+4β12−2β1+b1κβ22−2β12ϵcoth12β1x−λt+ζ0+12β1β2κ.(27) ).

Figure 6. (a,b) Solitary traveling wave solutions for Equation (Equation28(28) $\begin{aligned} u_{11} (x, t) & = \frac{1}{2} (\frac{2 β_{1} d_{2} {(η ϵ \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ)}^{2}}{{(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} {(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}} \\ + \frac{\sqrt{b_{1}^{2} β_{2}^{2} κ^{2} + 4 β_{1}^{2}} + b_{1} β_{2} κ - 2 β_{1} (β_{1} d_{2} κ + 1)}{β_{1} κ} - b_{1} \sqrt{\frac{β_{1}}{β_{3}}} (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1) \\ + (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1) β_{2} \sqrt{\frac{β_{1}}{β_{3}}} d_{2}) . \end{aligned}$ (28) ) and Equation (Equation29(29) $\begin{aligned} u_{12} (x, t) & = b_{1} (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1) + (ϵ^{2} (η \sqrt{p^{2} + 1} \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + 1 \\ {- p \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} β_{1} d_{2})/ ({(η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} \\ {(η \sqrt{p^{2} + 1} + p ϵ + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}) \\ - \frac{\sqrt{b_{1}^{2} β_{2}^{2} κ^{2} + 4 β_{1}^{2}} + b_{1} β_{2} κ - 2 β_{1} (β_{1} d_{2} κ + 1)}{2 β_{1} κ} + β_{2} d_{2} (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1) . \end{aligned}$ (29) ).

Figure 6. (a,b) Solitary traveling wave solutions for Equation (Equation28(28) u11(x,t)=122β1d2ηϵcoshβ1x−λt+ζ0+ϵ2η+coshβ1x−λt+ζ02η+coshβ1x−λt+ζ0+ϵsinhβ1x−λt+ζ02+b12β22κ2+4β12+b1β2κ−2β1β1d2κ+1β1κ−b1β1β3ϵsinhβ1x−λt+ζ0η+coshβ1x−λt+ζ0+1+ϵsinhβ1x−λt+ζ0η+coshβ1x−λt+ζ0+1β2β1β3d2.(28) ) and Equation (Equation29(29) u12(x,t)=b1−ϵp+sinhβ1x−λt+ζ0ηp2+1+coshβ1x−λt+ζ0−1+ϵ2ηp2+1coshβ1x−λt+ζ0+1−psinhβ1x−λt+ζ02β1d2ηp2+1+coshβ1x−λt+ζ02ηp2+1+pϵ+coshβ1x−λt+ζ0+ϵsinhβ1x−λt+ζ02−b12β22κ2+4β12+b1β2κ−2β1β1d2κ+12β1κ+β2d2−ϵp+sinhβ1x−λt+ζ0ηp2+1+coshβ1x−λt+ζ0−1.(29) ).

Figure 7. (a) Solitary wave solution for Equation (Equation31(31) $\begin{aligned} u_{13} (x, t) & = β_{1} d_{2} (ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1) + \frac{β_{1} d_{2} ϵ^{2} c s c h {[\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})]}^{4}}{4 {(ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}} \\ - \frac{2 β_{1} (\sqrt{β_{1}^{2} β_{2}^{2} ε^{2} κ (2 ε + 5 κ)} + 2 β_{1} β_{2} κ (ε + κ)) (ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}{β_{2}^{3} (4 κ^{3} - ε^{2} κ)} \\ - \frac{\sqrt{β_{1}^{2} β_{2}^{2} ε^{2} κ (2 ε + 5 κ)} + β_{1} β_{2} ε κ (β_{1} d_{2} (ε + 2 κ) + 1)}{β_{1} β_{2} ε κ (ε + 2 κ)} . \end{aligned}$ (31) ) and (b) solitary traveling wave solution for Equation (Equation32(32) $\begin{aligned} u_{14} (x, t) & = \frac{β_{1} d_{2} {(η ϵ \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ)}^{2}}{{(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} {(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}} \\ - \frac{\sqrt{β_{1}^{2} β_{2}^{2} ε^{2} κ (2 ε + 5 κ)} + β_{1} β_{2} ε κ (β_{1} d_{2} (ε + 2 κ) + 1)}{β_{1} β_{2} ε κ (ε + 2 κ)} \\ + \frac{1}{2} β_{2} \sqrt{\frac{β_{1}}{β_{3}}} d_{2} (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1) \\ - \frac{\sqrt{\frac{β_{1}}{β_{3}}} (\sqrt{β_{1}^{2} β_{2}^{2} ε^{2} κ (2 ε + 5 κ)} + 2 β_{1} β_{2} κ (ε + κ)) (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}{β_{2}^{2} (4 κ^{3} - ε^{2} κ)} . \end{aligned}$ (32) ).

Figure 7. (a) Solitary wave solution for Equation (Equation31(31) u13(x,t)=β1d2ϵcoth12β1x−λt+ζ0+1+β1d2ϵ2csch12β1x−λt+ζ044ϵcoth12β1x−λt+ζ0+12−2β1β12β22ε2κ(2ε+5κ)+2β1β2κ(ε+κ)ϵcoth12β1x−λt+ζ0+1β234κ3−ε2κ−β12β22ε2κ(2ε+5κ)+β1β2εκβ1d2(ε+2κ)+1β1β2εκ(ε+2κ).(31) ) and (b) solitary traveling wave solution for Equation (Equation32(32) u14(x,t)=β1d2ηϵcoshβ1x−λt+ζ0+ϵ2η+coshβ1x−λt+ζ02η+coshβ1x−λt+ζ0+ϵsinhβ1x−λt+ζ02−β12β22ε2κ(2ε+5κ)+β1β2εκβ1d2(ε+2κ)+1β1β2εκ(ε+2κ)+12β2β1β3d2ϵsinhβ1x−λt+ζ0η+coshβ1x−λt+ζ0+1−β1β3β12β22ε2κ(2ε+5κ)+2β1β2κ(ε+κ)ϵsinhβ1x−λt+ζ0η+coshβ1x−λt+ζ0+1β224κ3−ε2κ.(32) ).

Figure 8. (a) Solitary traveling wave solution for Equation (Equation33(33) $\begin{aligned} u_{15} (x, t) & = β_{1} d_{2} ϵ^{2} {(η \sqrt{p^{2} + 1} \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] - p \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}/ \\ ({(η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} (η \sqrt{p^{2} + 1} + p ϵ + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] \\ {+ ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}) β_{2} d_{2} (- (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1)) \\ + \frac{2 (\sqrt{β_{1}^{2} β_{2}^{2} ε^{2} κ (2 ε + 5 κ)} + 2 β_{1} β_{2} κ (ε + κ)) (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1)}{β_{2}^{2} (4 κ^{3} - ε^{2} κ)} \\ - \frac{\sqrt{β_{1}^{2} β_{2}^{2} ε^{2} κ (2 ε + 5 κ)} + β_{1} β_{2} ε κ (β_{1} d_{2} (ε + 2 κ) + 1)}{β_{1} β_{2} ε κ (ε + 2 κ)} . \end{aligned}$ (33) ) and (b) bright soliton solution for Equation (Equation36(36) $\begin{aligned} u_{17} (x, t) & = a_{0} + \frac{β_{1} d_{2} {(η ϵ \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ)}^{2}}{{(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} {(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}} \\ - \frac{1}{4} β_{1} d_{2} {(\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}^{2} + \frac{1}{2} β_{2} \sqrt{\frac{β_{1}}{β_{3}}} d_{2} (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1) . \end{aligned}$ (36) ).

Figure 8. (a) Solitary traveling wave solution for Equation (Equation33(33) u15(x,t)=β1d2ϵ2ηp2+1coshβ1x−λt+ζ0−psinhβ1x−λt+ζ0+12ηp2+1+coshβ1x−λt+ζ02ηp2+1+pϵ+coshβ1x−λt+ζ0+ϵsinhβ1x−λt+ζ02)β2d2−−ϵp+sinhβ1x−λt+ζ0ηp2+1+coshβ1x−λt+ζ0−1+2β12β22ε2κ(2ε+5κ)+2β1β2κ(ε+κ)−ϵp+sinhβ1x−λt+ζ0ηp2+1+coshβ1x−λt+ζ0−1β224κ3−ε2κ−β12β22ε2κ(2ε+5κ)+β1β2εκβ1d2(ε+2κ)+1β1β2εκ(ε+2κ).(33) ) and (b) bright soliton solution for Equation (Equation36(36) u17(x,t)=a0+β1d2ηϵcoshβ1x−λt+ζ0+ϵ2η+coshβ1x−λt+ζ02η+coshβ1x−λt+ζ0+ϵsinhβ1x−λt+ζ02−14β1d2ϵsinhβ1x−λt+ζ0η+coshβ1x−λt+ζ0+12+12β2β1β3d2ϵsinhβ1x−λt+ζ0η+coshβ1x−λt+ζ0+1.(36) ).

Figure 9. (a) Solitary wave solutions for Equation (Equation37(37) $\begin{aligned} u_{18} (x, t) & = a_{0} + β_{1} d_{2} ϵ^{2} {(η \sqrt{p^{2} + 1} \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] - p \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}/ \\ ({(η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} \\ {(η \sqrt{p^{2} + 1} + p ϵ + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}) \\ - β_{2} d_{2} (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1) \\ - β_{3} d_{2} {(\frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}^{2} . \end{aligned}$ (37) ) and (b) bright soliton solution for Equation (Equation39(39) $\begin{aligned} u_{19} (x, t) & = \frac{1}{4} (\frac{2}{ε} + \frac{2}{κ} - \frac{3}{ε + κ} - 4 β_{1} d_{2} + 4 β_{1} d_{2} (ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1) \\ - \frac{β_{1} d_{2} (ε^{2} - 4 κ^{2}) {(ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}}{ε^{2}} \\ + \frac{β_{1} d_{2} ϵ^{2} c s c h {[\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})]}^{4}}{{(ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}^{2}} - \frac{2 β_{1}^{2} (2 ε + κ) (ϵ \coth [\frac{1}{2} \sqrt{β_{1}} (x - λ t + ζ_{0})] + 1)}{β_{2}^{2} κ (ε + κ)}) . \end{aligned}$ (39) ).

Figure 9. (a) Solitary wave solutions for Equation (Equation37(37) u18(x,t)=a0+β1d2ϵ2ηp2+1coshβ1x−λt+ζ0−psinhβ1x−λt+ζ0+12ηp2+1+coshβ1x−λt+ζ02ηp2+1+pϵ+coshβ1x−λt+ζ0+ϵsinhβ1x−λt+ζ02−β2d2−ϵp+sinhβ1x−λt+ζ0ηp2+1+coshβ1x−λt+ζ0−1−β3d2ϵp+sinhβ1x−λt+ζ0ηp2+1+coshβ1x−λt+ζ0+12.(37) ) and (b) bright soliton solution for Equation (Equation39(39) u19(x,t)=142ε+2κ−3ε+κ−4β1d2+4β1d2ϵcoth12β1x−λt+ζ0+1−β1d2ε2−4κ2ϵcoth12β1x−λt+ζ0+12ε2+β1d2ϵ2csch12β1x−λt+ζ04ϵcoth12β1x−λt+ζ0+12−2β12(2ε+κ)ϵcoth12β1x−λt+ζ0+1β22κ(ε+κ).(39) ).

Figure 10. (a,b) Solitary wave solutions for Equation (Equation40(40) $\begin{aligned} u_{20} (x, t) & = \frac{1}{16} (\frac{16 β_{1} d_{2} {(η ϵ \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ)}^{2}}{{(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} {(η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}} \\ + 4 (- 4 β_{1} d_{2} - \frac{3}{ε + κ} + \frac{2}{ε} + \frac{2}{κ}) - \frac{4 β_{1} \sqrt{\frac{β_{1}}{β_{3}}} (2 ε + κ) (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}{β_{2} κ (ε + κ)} \\ + 8 β_{2} \sqrt{\frac{β_{1}}{β_{3}}} d_{2} (\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1) - \frac{β_{2}^{2} d_{2} (ε^{2} - 4 κ^{2}) {(\frac{ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})]}{η + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}^{2}}{β_{3} ε^{2}}) . \end{aligned}$ (40) ) and Equation (Equation41(41) $\begin{aligned} u_{21} (x, t) & = \frac{1}{4} (\frac{2}{ε} + \frac{2}{κ} - \frac{3}{ε + κ} - 4 β_{1} d_{2} + (4 ϵ^{2} (η \sqrt{p^{2} + 1} \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + 1 \\ {- p \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} β_{1} d_{2})/ ({(η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2} \\ {(η \sqrt{p^{2} + 1} + p ϵ + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})] + ϵ \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}^{2}) \\ + \frac{2 β_{1} (2 ε + κ) (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1)}{β_{2} κ (ε + κ)} - 4 (- \frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} - 1) \\ β_{2} d_{2} - \frac{β_{2}^{2} d_{2} (ε^{2} - 4 κ^{2}) {(\frac{ϵ (p + \sinh [\sqrt{β_{1}} (x - λ t + ζ_{0})])}{η \sqrt{p^{2} + 1} + \cosh [\sqrt{β_{1}} (x - λ t + ζ_{0})]} + 1)}^{2}}{β_{1} ε^{2}}) . \end{aligned}$ (41) ).

Nonlinear wave solutions of the Kudryashov–Sinelshchikov dynamical equation in mixtures liquid-gas bubbles under the consideration of heat transfer and viscosity

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Nonlinear wave solutions of the Kudryashov–Sinelshchikov dynamical equation in mixtures liquid-gas bubbles under the consideration of heat transfer and viscosity

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