1. Introduction
2. N-soliton solutions of the (2+1)-dimensional BLMP equation
3. A breather-soliton molecule and some related interaction solutions
Figure 1. Dynamical behaviors of the breather-soliton molecule. The parameters are given by k1r = 1.56, k1i = 0.25, k3 = 1.5, p1r = 0.5, p1i = 4.68, p3 = −0.25. |
Figure 2. Dynamical behaviors of the hybrid solution between a breather-soliton and a bell-shaped soliton. The parameters are chosen as ${k}_{1r}=\sqrt{7}$, k1i = 1, k3 = 2, k4 = 2, ${p}_{1r}=\tfrac{1}{2}$, ${p}_{1i}=\tfrac{3\sqrt{7}}{4}$, ${p}_{3}=-\tfrac{1}{4}$, p4 = 4. |
Figure 3. Dynamical behaviors of the hybrid solution mixed by a breather-soliton molecule and breather. |
Figure 4. The degeneration process of the solution mixed by a breather-soliton molecule and lump from {BSM, B}. The parameters are chosen as flows: k1r = 4, k1i = 2, k3 = 2, p1r = 0.17, p1i = 0.83, p3 = − 0.25, p4r = 0.03, p4i = 1.5. (a1)(b1) k4r = 1, k4i = 2; (a2)(b2) ${k}_{4r}=\tfrac{2}{5}$, k4i = 1; (a3)(b3) ${k}_{4r}=\tfrac{1}{5}$, ${k}_{4i}=\tfrac{1}{8}$. |
Figure 5. Dynamical behaviors of the interaction solution composed by a breather-soliton molecule, a breather and a soliton. The parameters are selected as k1r = 1.32, k1i = 0.25, k3 = 1.5, k4r = 1.73, k4i = 0.5, k6 = 1.5, p1r = 0.5, p1i = 3.17, p3 = −0.25, p4r = 0.33, p4i =2.02, p6 = 4. |
Figure 6. The degeneration process of the solution mixed by a breather-soliton molecule, a lump and a soliton from {BSM, B, S}. The parameters are chosen as flows: k1r = 1.732, k1i = 0.5, k3 = 1.5, p1r = 0.33, p1i = 2.02, p3 = −0.25, p4r = 0.08, p4i = 4, k6 = 1.5, p6 = 4. (a1)(b1) k4r = 0.8, k4i = 0.8; (a2)(b2) k4r = 0.5, k4i = 0.8; (a3)(b3) k4r = 0.02, k4i = 0.2. |
Figure 7. Dynamical behaviors of the interaction solution mixed by two different breather-soliton molecules. The parameters are given by k1r = 1.32, k1i = 0.25, k3 = 1.5, k4r = 1.73, k4i = 0.5, k6 = 1.25, p1r = 0.5, p1i = 3.17, p3 = −0.25, p4r = 0.33, p4i = 2.02, p6 = −0.1. |
Table 1. Breather-soliton molecule and interaction solutions. |
N = 3 + P | Name of the solution | Restriction conditions |
---|---|---|
P = 0 | {BSM} | ${k}_{1}={k}_{2}^{* }={k}_{1r}+{\rm{i}}{k}_{1i},\quad {p}_{1}={p}_{2}^{* }\,=\,{p}_{1r}+{\rm{i}}{p}_{1i}$, |
$\tfrac{{k}_{1r}}{{k}_{3}}=\tfrac{{k}_{1r}\ {p}_{1r}-{k}_{1i}\ {p}_{1i}}{{k}_{3}\ {p}_{3}}=\tfrac{{k}_{1r}^{3}-3{k}_{1r}\ {k}_{1i}^{2}}{{k}_{3}^{3}}$ | ||
P = 1 | {BSM, S} | the same as above |
P = 2 | {BSM, B} | ${k}_{1}={k}_{2}^{* }={k}_{1r}+{\rm{i}}{k}_{1i},\quad {p}_{1}={p}_{2}^{* }\,=\,{p}_{1r}+{\rm{i}}{p}_{1i}$, |
${k}_{4}={k}_{5}^{* }={k}_{4r}+{\rm{i}}{k}_{4i},\quad {p}_{4}={p}_{5}^{* }={p}_{4r}+{\rm{i}}{p}_{4i}$ | ||
$\tfrac{{k}_{1r}}{{k}_{3}}=\tfrac{{k}_{1r}\ {p}_{1r}-{k}_{1i}\ {p}_{1i}}{{k}_{3}\ {p}_{3}}=\tfrac{{k}_{1r}^{3}-3{k}_{1r}\ {k}_{1i}^{2}}{{k}_{3}^{3}}$, | ||
P = 2 | {BSM, L} | ${k}_{1}={k}_{2}^{* }={k}_{1r}+{\rm{i}}{k}_{1i},\quad {p}_{1}={p}_{2}^{* }\,=\,{p}_{1r}+{\rm{i}}{p}_{1i}$, |
${k}_{4}={k}_{5}^{* }={k}_{4r}+{\rm{i}}{k}_{4i},\quad {p}_{4}={p}_{5}^{* }={p}_{4r}+{\rm{i}}{p}_{4i}$, | ||
$\tfrac{{k}_{1r}}{{k}_{3}}=\tfrac{{k}_{1r}\ {p}_{1r}-{k}_{1i}\ {p}_{1i}}{{k}_{3}\ {p}_{3}}=\tfrac{{k}_{1r}^{3}-3{k}_{1r}\ {k}_{1i}^{2}}{{k}_{3}^{3}}$, | ||
${k}_{4r}^{2}+{k}_{4i}^{2}\to 0$ | ||
P = 3 | {BSM, B, S} | the same as what's used in {BSM, B} |
P = 3 | {BSM, L, S} | the same as what's used in {BSM, L} |
P = 3 | {BSM, BSM} | ${k}_{1}={k}_{2}^{* }={k}_{1r}+{\rm{i}}{k}_{1i},\quad {p}_{1}={p}_{2}^{* }\,=\,{p}_{1r}+{\rm{i}}{p}_{1i}$, |
${k}_{4}={k}_{5}^{* }={k}_{4r}+{\rm{i}}{k}_{4i},\quad {p}_{4}={p}_{5}^{* }\,=\,{p}_{4r}+{\rm{i}}{p}_{4i}$, | ||
$\tfrac{{k}_{1r}}{{k}_{3}}=\tfrac{{k}_{1r}\ {p}_{1r}-{k}_{1i}\ {p}_{1i}}{{k}_{3}\ {p}_{3}}=\tfrac{{k}_{1r}^{3}-3{k}_{1r}\ {k}_{1i}^{2}}\ {{k}_{3}^{3}}$, | ||
$\tfrac{{k}_{4r}}{{k}_{6}}=\tfrac{{k}_{4r}\ {p}_{4r}-{k}_{4i}\ {p}_{4i}}{{k}_{6}\ {p}_{6}}=\tfrac{{k}_{4r}^{3}-3{k}_{4r}\ {k}_{4i}^{2}}{{k}_{6}^{3}}$ |
4. A breather molecule and some related interaction solutions
Figure 8. Dynamical behaviors of the breather molecule. The parameters are set as k1r = 1.73, k1i = 1.25, k3r = 1.39, k3i = 1.10, p1r = 0.73, p1i = 3, p3r = 0.73 and p3i = 2.73. |
Figure 9. Dynamical behaviors of the mixed solution by a breather molecule and soliton. The parameters are set as k1r = 1.73, k1i = 1.25, k3r = 1.39, k3i = 1.10, k5 = 2, p1r = 0.73, p1i = 3, p3r = 0.73 , p3i = 2.73 and p5 = 3.23. |
Figure 10. Dynamical behaviors of the interaction solution between a breather molecule and breather. The parameters are chosen as: k1 = 1.73 + 1.25i, k3 = 1.39 + 1.10i, k5 = 1.04 + 0.6i, p1 = 0.73 + 3i, p3 = 0.73 + 2.73i, p5 = 0.73 + 2.73i. |
Figure 11. Dynamical behaviors of a breather-breather-breather molecule. The parameters are chosen as:k1 = 1.73 + 1.25i, k3 = 1.39 + 1.10i, k5 = 1.04 + 0.96i, p1 = 0.73 + 3i, p3 = 0.73 + 1.73i, p5 = 0.73 + 2.34i. |
Figure 12. The degeneration process of the solution mixed by a breather molecule and lump from {BB, B}. The parameters are chosen as flows: k1r = 1.73, k1i = 1.25, k3r = 1.39, k3i = 1.10, p1r = 0.73, p1i = 3, p3r = 0.73, p3i = 2.73, p5r = 0.15, p5i = 1.5. (a1)(b1) k5r = 1, k5i = 0.6; (a2)(b2) k5r = 0.5, k5i = 0.4; (a3)(b3) k5r = 0.1, k5i = 0.3. |
Table 2. Breather molecule and interaction solutions. |
N = 4 + P | Name of the solution | Restriction conditions |
---|---|---|
P = 0 | {BBM} | ${k}_{1}={k}_{2}^{* }={k}_{1r}+{\rm{i}}{k}_{1i},\quad {p}_{1}={p}_{2}^{* }\,=\,{p}_{1r}+{\rm{i}}{p}_{1i}$, |
${k}_{3}={k}_{4}^{* }={k}_{3r}+{\rm{i}}{k}_{3i},\quad {p}_{3}={p}_{4}^{* }={p}_{3r}+{\rm{i}}{p}_{3i}$, | ||
$\tfrac{{k}_{1r}}{{k}_{3r}}=\tfrac{{k}_{1r}\ {p}_{1r}-{k}_{1i}\ {p}_{1i}}{{k}_{3r}\ {p}_{3r}-{k}_{3i}\ {p}_{3i}}=\tfrac{{k}_{1r}^{3}-3{k}_{1r}\ {k}_{1i}^{2}}{{k}_{3r}^{3}-3{k}_{3r}\ {k}_{3i}^{2}}$ | ||
P = 1 | {BBM, S} | the same as above |
P = 2 | {BBM, B} | ${k}_{1}={k}_{2}^{* }={k}_{1r}+{\rm{i}}{k}_{1i},\quad {p}_{1}={p}_{2}^{* }\,=\,{p}_{1r}+{\rm{i}}{p}_{1i}$, |
${k}_{3}={k}_{4}^{* }={k}_{3r}+{\rm{i}}{k}_{3i},\quad {p}_{3}={p}_{4}^{* }={p}_{3r}+{\rm{i}}{p}_{3i}$, | ||
$\tfrac{{k}_{1r}}{{k}_{3r}}=\tfrac{{k}_{1r}\ {p}_{1r}-{k}_{1i}\ {p}_{1i}}{{k}_{3r}\ {p}_{3r}-{k}_{3i}\ {p}_{3i}}=\tfrac{{k}_{1r}^{3}-3{k}_{1r}\ {k}_{1i}^{2}}{{k}_{3r}^{3}-3{k}_{3r}\ {k}_{3i}^{2}}$, | ||
${k}_{5}={k}_{6}^{* }={k}_{5r}+{\rm{i}}{k}_{5i},\quad {p}_{5}={p}_{6}^{* }={p}_{5r}+{\rm{i}}{p}_{5i}$ | ||
P = 2 | {BBM, L} | ${k}_{1}={k}_{2}^{* }={k}_{1r}+{\rm{i}}{k}_{1i},\quad {p}_{1}={p}_{2}^{* }\,=\,{p}_{1r}+{\rm{i}}{p}_{1i}$, |
${k}_{3}={k}_{4}^{* }={k}_{3r}+{\rm{i}}{k}_{3i},\quad {p}_{3}={p}_{4}^{* }={p}_{3r}+{\rm{i}}{p}_{3i}$, | ||
${k}_{5}={k}_{6}^{* }={k}_{5r}+{\rm{i}}{k}_{5i},\quad {p}_{5}={p}_{6}^{* }={p}_{5r}+{\rm{i}}{p}_{5i},$ | ||
$\tfrac{{k}_{1r}}{{k}_{3r}}=\tfrac{{k}_{1r}\ {p}_{1r}-{k}_{1i}\ {p}_{1i}}{{k}_{3r}\ {p}_{3r}-{k}_{3i}\ {p}_{3i}}=\tfrac{{k}_{1r}^{3}-3{k}_{1r}\ {k}_{1i}^{2}}{{k}_{3r}^{3}-3{k}_{3r}\ {k}_{3i}^{2}}$, | ||
${k}_{5r}^{2}+{k}_{5i}^{2}\to 0$ | ||
P = 2 | {BBBM} | ${k}_{1}={k}_{2}^{* }={k}_{1r}+{\rm{i}}{k}_{1i},\quad {p}_{1}={p}_{2}^{* }\,=\,{p}_{1r}+{\rm{i}}{p}_{1i}$, |
${k}_{3}={k}_{4}^{* }={k}_{3r}+{\rm{i}}{k}_{3i},\quad {p}_{3}={p}_{4}^{* }={p}_{3r}+{\rm{i}}{p}_{3i}$, | ||
${k}_{5}={k}_{6}^{* }={k}_{5r}+{\rm{i}}{k}_{5i},\quad {p}_{5}={p}_{6}^{* }={p}_{5r}+{\rm{i}}{p}_{5i}$, | ||
$\tfrac{{k}_{1r}}{{k}_{3r}}=\tfrac{{k}_{1r}\ {p}_{1r}-{k}_{1i}\ {p}_{1i}}{{k}_{3r}\ {p}_{3r}-{k}_{3i}\ {p}_{3i}}=\tfrac{{k}_{1r}^{3}-3{k}_{1r}\ {k}_{1i}^{2}}{{k}_{3r}^{3}-3{k}_{3r}\ {k}_{3i}^{2}}$, | ||
$\tfrac{{k}_{1r}}{{k}_{5r}}=\tfrac{{k}_{1r}\ {p}_{1r}-{k}_{1i}\ {p}_{1i}}{{k}_{5r}\ {p}_{5r}-{k}_{5i}\ {p}_{5i}}=\tfrac{{k}_{1r}^{3}-3{k}_{1r}\ {k}_{1i}^{2}}{{k}_{5r}^{3}-3{k}_{5r}\ {k}_{5i}^{2}}$ |