Theory and Modern Applications
From: Conservation laws and exact solutions of the \((3+1)\)-dimensional Jimbo–Miwa equation
[⋅,⋅] | \(X_{1}\) | \(X_{2}\) | \(X_{3}\) | \(X_{4}\) | \(X_{5}^{f}\) | \(X_{6}^{g}\) | \(X_{7}^{h}\) | \(X_{8}^{d}\) |
---|---|---|---|---|---|---|---|---|
\(X_{1}\) | 0 | \(-3X_{2}\) | 0 | 0 | \(2X_{5}^{f}\) | \(X_{6}^{3t g'-g}\) | \(-X_{7}^{h}\) | \(X_{8}^{3t d_{t}+d}\) |
\(X_{2}\) | \(3X_{2}\) | 0 | 0 | 0 | \(X_{7}^{3f'/4}\) | \(X_{6}^{g'}\) | 0 | \(X_{8}^{d_{t}}\) |
\(X_{3}\) | 0 | 0 | 0 | \(-X_{4}\) | \(X_{5}^{zf'-f}\) | 0 | \(X_{7}^{zh'}\) | \(X_{8}^{zd_{z}}\) |
\(X_{4}\) | 0 | 0 | \(X_{4}\) | 0 | \(X_{5}^{f'}\) | 0 | \(X_{7}^{h'}\) | \(X_{8}^{d_{z}}\) |
\(X_{5}^{\tilde{f}}\) | \(-2X_{5}^{\tilde{f}}\) | \(-X_{7}^{3\tilde{f}'/4}\) | \(-X_{5}^{z\tilde{f}'-\tilde{f}}\) | \(-X_{5}^{\tilde{f}'}\) | \(X_{5}^{3t(f\tilde{f}''-\tilde{f}f'')/4}\) | \(X_{8}\) | \(X_{8}\) | 0 |
\(X_{6}^{\tilde{g}}\) | \(-X_{6}^{3t \tilde{g}'-\tilde{g}}\) | \(-X_{6}^{\tilde{g}'}\) | 0 | 0 | \(-X_{8}\) | \(X_{8}\) | \(X_{8}\) | 0 |
\(X_{7}^{\tilde{h}}\) | \(X_{7}^{\tilde{h}}\) | 0 | \(-X_{7}^{z\tilde{h}'}\) | \(-X_{7}^{\tilde{h}'}\) | \(-X_{8}\) | \(-X_{8}\) | 0 | 0 |
\(X_{8}^{\tilde{d}}\) | \(-X_{8}^{3t \tilde{d}_{t}+\tilde{d}}\) | \(-X_{8}^{\tilde{d}_{t}}\) | \(-X_{8}^{z\tilde{d}_{z}}\) | \(-X_{8}^{\tilde{d}_{z}}\) | 0 | 0 | 0 | 0 |