1. Introduction
2. Riccati Bernoulli sub-ode method
2.1. Bäcklund transformation
3. Applications
3.1. Applications of Riccati-Bernoulli sub-ode method to Eq. (1.1)
Fig. 1. Graphs of the real part of solutions $\psi_{1,1}(x, t) $ and $\psi_{1,2}(x, t) $ to Eq. (1.1), respectively, for the values k=3,α=−0.5, a3=3 and K=4. |
Fig. 2. Graphs of the solutions $\psi_{1,3}(x, t) $ and $\psi_{1,4}(x, t) $ of Eq. (1.1), respectively, for the values k=3, α=−0.5, a3=3 and K=4. |
Fig. 3. Graphs of the real part of solutions ψ2,1(x, t) and ψ2,2(x, t) to Eq. (1.1), respectively, for the values k=1, v=3, a3=1 and K=1. |
Fig. 4. Graphs of the solutions ψ2,3(x,t) and ψ2,4(x,t)of Eq. (1.1), respectively, for the values k=3,v=1,a3=3 and K=1. |
Fig. 5. Graphs of the real part of solutions ${{\psi }_{3,1}}\left( x,t \right)$ and ${{\psi }_{3,2}}\left( x,t \right)$ to Eq. (1.1), respectively, for the values α=0.5, v=5, a3=3 and K=1. |
Fig. 6. Graphs of the solutions ${{\psi }_{3,3}}\left( x,t \right)$ and ${{\psi }_{3,4}}\left( x,t \right)$ of Eq.(1.1), respectively, for the values α=0.25, v=−1, a3=1 and K=1. |
Fig. 7. Graphs of the solutions ${{\psi }_{4,1}}\left( x,t \right)$ and ${{\psi }_{4,2}}\left( x,t \right)$ of Eq. (1.1), respectively, for the values α=0.5, v=1, a3=1 and K=1. |
Fig. 8. Graphs of the solutions ${{\psi }_{4,3}}\left( x,t \right)$ and ${{\psi }_{4,4}}\left( x,t \right)$ of Eq. (1.1), respectively, for the values α=0.5, v=−1, a3=1 and K=1. |
Fig. 9. Graphs of the solutions ${{\psi }_{5,1}}\left( x,t \right)$ and ${{\psi }_{5,2}}\left( x,t \right)$ of Eq. (1.1), respectively, for the values k=−2, v=1, a3=3, K=4. |
Fig. 10. Graphs of the solutions ${{\psi }_{5,3}}\left( x,t \right)$ and ${{\psi }_{5,4}}\left( x,t \right)$ of Eq. (1.1), respectively, for the values k=3, v=−3, a3=3 and K=−2. |
Fig. 11. Graphs of the solutions ${{\psi }_{6,1}}\left( x,t \right)$ and ${{\psi }_{6,2}}\left( x,t \right)$ of Eq. (1.1), respectively, for the values α=−0.5, k=1, a3=2 and K=4. |
Fig. 12. Graphs of the solutions ${{\psi }_{6,3}}\left( x,t \right)$ and ${{\psi }_{6,4}}\left( x,t \right)$ of Eq. (1.1), respectively, for the values α=0.5, k=−1, a3=−1 and K=3. |
Fig.13. Graphs of the solutions $\psi_{1,1}(x, t) $ and $\psi_{1,2}(x, t) $ of Eq. (1.2), respectively, for the values k=1, α=−0.5, a3=0.5, a1=1, β=−1 and K=1. |
Fig. 14. Graph of the solution \psi_{2,1}(x, t) of Eq. (1.2) for the values α=−0.5, β=0.5, a1=−1, k=3 and K=3. |