1. Introduction
2. Symmetry analysis for coupled equation
3. Optimal system
Table 1. Commutator table for the weakly coupled B-KP Eq. (3). |
[Vi,Vj] | V1 | V2 | V3 | V4 | V5 | V6 | V7 | V8 |
---|---|---|---|---|---|---|---|---|
V1 | 0 | −V2 | $\frac{-V_{3}}{3}$ | $\frac{2}{3} V_{4}$ | $-\frac{V_{5}}{3}$ | 0 | $\mathrm{v}_{7}\left(t f_{1}^{\prime}\right) $ | $\mathrm{V}_{8}\left(t f_{2}^{\prime}+\frac{f_{2}}{3}\right) $ |
V2 | V2 | 0 | 0 | V5 | 0 | 0 | $\mathrm{v}_{7}\left(f_{1}^{\prime}\right) $ | $\mathrm{V}_{8}\left(f_{2}^{\prime}\right) $ |
V3 | $\frac{V_{3}}{3}$ | 0 | 0 | $\frac{2}{\alpha} \mathrm{V}_{8}$ | 0 | 0 | 0 | 0 |
V4 | $-\frac{2}{3} V_{4}$ | −V5 | $-\frac{2}{\alpha} \mathrm{V}_{8}$ | 0 | $-\frac{1}{\alpha} \mathrm{V}_{8}$ | 0 | 0 | 0 |
V5 | $\frac{V_{5}}{3}$ | 0 | 0 | $\frac{V_{8}}{\alpha}$ | 0 | 0 | 0 | 0 |
V6 | 0 | 0 | 0 | 0 | 0 | 0 | −V7 | 0 |
V7 | $-\mathrm{V}_{7}\left(t f_{1}^{\prime}\right) $ | $-\mathrm{v}_{7}\left(f_{1}^{\prime}\right) $ | $0 | 0 | 0 | V7 | 0 | 0 |
V8 | $-\mathrm{V}_{8}\left(t f_{2}^{\prime}+\frac{f_{2}}{3}\right) $ | $-\mathrm{v}_{8}\left(f_{2}^{\prime}\right) $ | 0 | 0 | 0 | 0 | 0 | 0 |
Table 2. Adjoint table for the weakly coupled B-KP Eq. (3). |
[Vi,Vj] | V1 | V2 | V3 | V4 | V5 | V6 | V7 | V8 |
---|---|---|---|---|---|---|---|---|
V1 | V1 | $e^{\epsilon} V_{2}$ | $e^{\frac{\epsilon}{3}} V_{3}$ | $e^{\frac{-2 \epsilon}{3}} \mathrm{~V}_{4}$ | $e^{\frac{\epsilon}{3}} \mathrm{~V}_{5}$ | V6 | $e^{-\epsilon} \mathrm{V}_{7}$ | $e^{-\epsilon} \mathrm{V}_{8}$ |
V2 | V1−ϵV2 | V2 | V3 | V4−ϵV5 | V5 | V6 | $e^{-\epsilon} \mathrm{V}_{7}$ | $e^{-\epsilon} \mathrm{V}_{8}$ |
V3 | $V_{1}-\frac{\epsilon}{3} V_{3}V3$ | V2 | V3 | $\mathrm{V}_{4}-\frac{2 \epsilon}{\alpha} \mathrm{V}_{8}$ | V5 | V6 | V7 | V8 |
V4 | $\mathrm{V}_{1}+\frac{2 \epsilon}{3} \mathrm{~V}_{4}$ | V2+ϵV5 | $\mathrm{V}_{3}+\frac{2 \epsilon}{\alpha} \mathrm{V}_{8}$ | V4 | $\mathrm{V}_{5}+\frac{\epsilon}{\alpha} \mathrm{V}_{8}$ | V6 | V7 | V8 |
V5 | $V_{1}-\frac{\epsilon}{3} V_{5}$ | V2 | V3 | $\mathrm{V}_{4}-\frac{\epsilon}{\alpha} \mathrm{V}_{8}$ | V5 | V6 | V7 | V8 |
V6 | V1 | V2 | V3 | V4 | V5 | V6 | $e^{\epsilon} \mathrm{V}_{7}$ | V8 |
V7 | V1+ϵV7 | V2+ϵV7 | V3 | V4 | V5 | $V_{6}-\epsilon V_{7}$ | V7 | V8 |
V8 | V1+ϵV8 | V2+ϵV8 | V3 | V4 | V5 | V6 | V7 | V8 |
4. Lie symmetry analysis and reduction of B-KP equation
5. Graphical illustrations
Fig. 1. Solitary wave form for Eq. (36). Figures (a), (d) indicate doubly soliton and multi soliton structures, (b), (e) represent corresponding contour plot while figures (c), (f) show wave propagation pattern. |
Fig. 2. Wave profile structures for Eq. (43). Figures (a), (c) exhibit singular-form and kink wave structures, while figures (b), (d) illustrate the propagation of waves along the x axis. |
Fig. 3. Annihilation of solitary wave form for Eq. (65). Figures (a), (d) reveal singularity form and soliton structures, (b), (e) represent corresponding annihilation while figures (c), (f) show wave propagation pattern along x axis. |
Fig. 4. Wave profile structures for Eq. (77). Figures (a), (c) exhibit multisoliton and wave structures, while figures (b), (d) represent wave propagation along x axis. |
Fig. 5. Singular wave-form structure for Eq. (85). Figures (a), (d) display various wave structures, (b), (e) represent corresponding contour plot while figures (c), (f) illustrate the propagation of waves along the x axis. |