| Abstract: |
| This talk aims to explore the relations between the Sylvester equation and semi-discrete integrable systems. Starting from the Sylvester equation $KM+ML=rs^\top$, the master function $S^{(i,j)}=s^\top L^jC(I+MC)^{-1}K^ir$ is introduced. By imposing dispersion relations on $r$ and $s$, the semi-discrete Toda equation, the modified semi-discrete Toda equation and their Miura transformation are established through equations of $S^{(i,j)}$. In addition, Lax pair and solutions are constructed for the semi-discrete Toda equation in a systematic way. Under the symmetric constraint $S^{(i,j)}=S^{(j,i)}$, the semi-discrete sine-Gordon equation, the modified semi-discrete sine-Gordon equation and their Miura transformation are derived. Integrability such as Lax pair, the bilinear form and various types of solutions for the semi-discrete sine-Gordon equation are presented as well. |
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