Theory and Modern Applications
Table 3 Variable separation solution of the LLG equation without damping (see equation (59)). F is the solution of (71); \(F_{2}\) satisfies (99)
Solutions I–II | Decay rate |
|---|---|
Solution I: \(\begin{pmatrix} {\frac{{2 \mathrm{e}^{F}}}{1+{\mathrm{e}^{2 F}}} \cos (C_{{1}}+ C_{{2}}\int {r}^{-n+1} ( {\mathrm{e}^{2 F}}+1 )^{2} {\mathrm{e}^{-2 F}} \,\mathrm{d}r + C_{5}t ) } \\ {\frac{{2 \mathrm{e}^{F}}}{1+{\mathrm{e}^{2 F}}} \sin (C_{{1}}+ C_{{2}}\int {r}^{-n+1} ( {\mathrm{e}^{2 F}}+1 )^{2} {\mathrm{e}^{-2 F}} \,\mathrm{d}r + C_{5}t ) } \\ {\frac{1-{\mathrm{e}^{2 F}}}{1+{\mathrm{e}^{2 F}}}} \end{pmatrix}\) | O(1) |
Solution II: \(\begin{pmatrix} {\frac{{2 \mathrm{e}^{F_{2}}}}{1+{\mathrm{e}^{2 F_{2}}}} \cos (C_{{1}}+ C_{{2}}\int ( {\mathrm{e}^{2 F_{2}}}+1 )^{2} {\mathrm{e}^{-2 F_{2}}} \,\mathrm{d}\overline{r} + C_{5}t ) } \\ {\frac{{2 \mathrm{e}^{F_{2}}}}{1+{\mathrm{e}^{2 F_{2}}}} \sin (C_{{1}}+ C_{{2}}\int ( {\mathrm{e}^{2 F_{2}}}+1 )^{2} {\mathrm{e}^{-2 F_{2}}}\, \mathrm{d}\overline{r} + C_{5}t ) } \\ {\frac{1-{\mathrm{e}^{2 F_{2}}}}{1+{\mathrm{e}^{2 F_{2}}}}} \end{pmatrix} \) | O(1) |