The Sylvester equation and integrable equations: I. The Korteweg-de Vries system and sine-Gordon equation

DOI

10.1080/14029251.2014.936759How to use a DOI?

Keywords

The Sylvester equation; integrable systems; Cauchy matrix approach; solutions

Abstract

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 KM + MK = r s^T we introduce a scalar function S^{(i, j)} = s^T K^j (I + M)⁻¹Kⁱr 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 r and s, 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^{(j, i)}. The solution M provides t function by t = ∣I + M∣. 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.

Copyright

© 2014 The Authors. Published by Atlantis Press and Taylor & Francis

Open Access

This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - Dan-dan Xu
AU  - Da-jun Zhang
AU  - Song-lin Zhao
PY  - 2021
DA  - 2021/01/06
TI  - The Sylvester equation and integrable equations: I. The Korteweg-de Vries system and sine-Gordon equation
JO  - Journal of Nonlinear Mathematical Physics
SP  - 382
EP  - 406
VL  - 21
IS  - 3
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2014.936759
DO  - 10.1080/14029251.2014.936759
ID  - Xu2021
ER  -