Breather molecules and localized interaction solutions in the (2+1)-dimensional BLMP equation

Jiaxin Qi,Hongli An,Peng Jin

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} ${k}_{1}={k}_{2}^{* }={k}_{1r}+{\rm{i}}{k}_{1i},\quad {p}_{1}={p}_{2}^{* }\,=\,{p}_{1r}+{\rm{i}}{p}_{1i}$, ${k}_{3}={k}_{4}^{* }={k}_{3r}+{\rm{i}}{k}_{3i},\quad {p}_{3}={p}_{4}^{* }={p}_{3r}+{\rm{i}}{p}_{3i}$, ${k}_{5}={k}_{6}^{* }={k}_{5r}+{\rm{i}}{k}_{5i},\quad {p}_{5}={p}_{6}^{* }={p}_{5r}+{\rm{i}}{p}_{5i}$, $\tfrac{{k}_{1r}}{{k}_{3r}}=\tfrac{{k}_{1r}\ {p}_{1r}-{k}_{1i}\ {p}_{1i}}{{k}_{3r}\ {p}_{3r}-{k}_{3i}\ {p}_{3i}}=\tfrac{{k}_{1r}^{3}-3{k}_{1r}\ {k}_{1i}^{2}}{{k}_{3r}^{3}-3{k}_{3r}\ {k}_{3i}^{2}}$, $\tfrac{{k}_{1r}}{{k}_{5r}}=\tfrac{{k}_{1r}\ {p}_{1r}-{k}_{1i}\ {p}_{1i}}{{k}_{5r}\ {p}_{5r}-{k}_{5i}\ {p}_{5i}}=\tfrac{{k}_{1r}^{3}-3{k}_{1r}\ {k}_{1i}^{2}}{{k}_{5r}^{3}-3{k}_{5r}\ {k}_{5i}^{2}}$