Theory and Modern Applications
Table 2 Variable separation solution, inhomogeneity terms, and decay rate of equivalent ILLG equation (27)
Solutions I–III | q(r) and decay rate |
|---|---|
Solution I: \(A \exp [i \Theta^{'}_{1} ]\), | \({\frac {C_{{1}}{r}^{-2 n+2} ( C_{{2}}{r}^{n}+C_{{3}}n ) ^{2}{\beta}^{2} ( {A}^{2}-1 ) }{\alpha ( {A}^{2}+1 ) C_{{2}}^{2}{n}^{2}}}\) |
\(\Theta^{'}_{1} = C_{1}t + {\frac{\alpha ( {A}^{2}+1 ) }{\beta ( {A}^{2}-1 ) }\ln ( {\frac{\beta ( {A}^{2}-1 ) ( C_{{2}}{r}^{n}+C_{{3}}n ) }{n\alpha ( {A}^{2}+1 ) }} ) }\) | and O(1) |
Solution II: \({\frac{\sin ( B ) }{1+\cos ( B ) }} \exp [i \Theta_{2} ]\), | \(-{\frac{C_{{1}}\cos ( B ) {\beta}^{2} ( \overline{r} +K_{{n+1}} ) ^{2}}{\alpha ( \overrightarrow{K} \cdot \overrightarrow{K} ) }}\) |
\(\Theta_{2} = C_{{1}}t -{\frac{\alpha\ln ( \overline{r} +K_{{n+1}} ) }{\cos ( B ) \beta}}+K_{{n+2}} \) | and O(1) |