1. Introduction
2. Derivation of the (2+1)-dimensional coupled Boussinesq equations
3. Conservation laws of the (2+1)-dimensional coupled Boussinesq equations
3.1. Lie symmetry analysis
3.2. Conservation laws
4. Exact solutions of the (2+1)-dimensional coupled Boussinesq equations
i | i. Setting ${C}_{1}=0$ and ${C}_{2}\ne 0$, we have $\begin{eqnarray*}\begin{array}{l}A={a}_{0}-\lambda \sqrt{-\displaystyle \frac{{\alpha }_{16}}{2{\alpha }_{14}}}+\sqrt{-\displaystyle \frac{{\alpha }_{16}({\lambda }^{2}-4\mu )}{{\alpha }_{14}}}\\ \tanh \left(\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}\zeta \right),\\ B={b}_{0}-\lambda \sqrt{-\displaystyle \frac{{\alpha }_{26}}{2{\alpha }_{24}}}+\sqrt{-\displaystyle \frac{{\alpha }_{26}({\lambda }^{2}-4\mu )}{{\alpha }_{24}}}\\ \tanh \left(\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}\zeta \right).\end{array}\end{eqnarray*}$ |
ii | ii. Setting ${C}_{1}\ne 0$ and ${C}_{2}=0$, we have $\begin{eqnarray*}\begin{array}{l}A={a}_{0}-\lambda \sqrt{-\displaystyle \frac{{\alpha }_{16}}{2{\alpha }_{14}}}+\sqrt{-\displaystyle \frac{{\alpha }_{16}({\lambda }^{2}-4\mu )}{{\alpha }_{14}}}\\ \coth \left(\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}\zeta \right),\\ B={b}_{0}-\lambda \sqrt{-\displaystyle \frac{{\alpha }_{26}}{2{\alpha }_{24}}}+\sqrt{-\displaystyle \frac{{\alpha }_{26}({\lambda }^{2}-4\mu )}{{\alpha }_{24}}}\\ \coth \left(\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}\zeta \right).\end{array}\end{eqnarray*}$ |
iii | iii. Setting ${C}_{1}\ne 0$ and ${C}_{1}^{2}\lt {C}_{2}^{2}$, we have $\begin{eqnarray*}\begin{array}{l}A={a}_{0}-\lambda \sqrt{-\displaystyle \frac{{\alpha }_{16}}{2{\alpha }_{14}}}+\sqrt{-\displaystyle \frac{{\alpha }_{16}({\lambda }^{2}-4\mu )}{{\alpha }_{14}}}\\ \tanh \left(\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}\zeta +{\theta }_{0}\right),\\ B={b}_{0}-\lambda \sqrt{-\displaystyle \frac{{\alpha }_{26}}{2{\alpha }_{24}}}+\sqrt{-\displaystyle \frac{{\alpha }_{26}({\lambda }^{2}-4\mu )}{{\alpha }_{24}}}\\ \tanh \left(\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}\zeta +{\theta }_{0}\right),\end{array}\end{eqnarray*}$ where ${\theta }_{0}={\tanh }^{-1}\left(\tfrac{{C}_{1}}{{C}_{2}}\right)$. |
iv | iv. Setting ${C}_{1}\ne 0$ and ${C}_{1}^{2}\gt {C}_{2}^{2}$, we have $\begin{eqnarray*}\begin{array}{l}A={a}_{0}-\lambda \sqrt{-\displaystyle \frac{{\alpha }_{16}}{2{\alpha }_{14}}}+\sqrt{-\displaystyle \frac{{\alpha }_{16}({\lambda }^{2}-4\mu )}{{\alpha }_{14}}}\\ \coth \left(\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}\zeta +{\theta }_{0}\right),\\ B={b}_{0}-\lambda \sqrt{-\displaystyle \frac{{\alpha }_{26}}{2{\alpha }_{24}}}+\sqrt{-\displaystyle \frac{{\alpha }_{26}({\lambda }^{2}-4\mu )}{{\alpha }_{24}}}\\ \coth \left(\displaystyle \frac{\sqrt{{\lambda }^{2}-4\mu }}{2}\zeta +{\theta }_{0}\right),\end{array}\end{eqnarray*}$ where ${\theta }_{0}={\tanh }^{-1}\left(\tfrac{{C}_{2}}{{C}_{1}}\right)$. |
Figure 1. Plot of solution A in equations ( |
Figure 2. Plot of solution A in equations ( |