2014/01/31 by Dan-dan Xu, Da-jun Zhang, Songling Zhao · 2 citations
Physics and Astronomy · #nlin.SI
published as J. Nonlin. Math. Phys., 21 (2014), 382-406 · 23 pages
arxiv created 2014/01/31 · arxiv updated 2014/08/28
The paper is to reveal the direct links between the well known Sylvester equation in matrix theory and some integrable systems. Using the Sylvester equation \boldsymbolK \boldsymbolM+\boldsymbolM \boldsymbolK=\boldsymbolr \boldsymbolsT we introduce a scalar function S(i,j)=\boldsymbolsT \boldsymbolKj(\boldsymbolI+\boldsymbolM)-1\boldsymbolKi\boldsymbolr which is defined as same as in discrete case. S(i,j) satisfy some recurrence relations which can be viewed as discrete equations and play indispensable roles in deriving continuous integrable equations. By imposing dispersion relations on \boldsymbolr and \boldsymbols, we find the Korteweg-de Vries equation, modified Korteweg-de Vries equation, Schwarzian Korteweg-de Vries equation and sine-Gordon equation can be expressed by some discrete equations of S(i,j) defined on certain points. Some special matrices are used to solve the Sylvester equation and prove symmetry property S(i,j)=S(i,j). The solution \boldsymbolM provides τ function by τ=|\boldsymbolI+\boldsymbolM|. We hope our results can not only unify the Cauchy matrix approach in both continuous and discrete cases, but also bring more links for integrable systems and variety of areas where the Sylvester equation appears frequently.