T R Vishnu
Articles written in Resonance – Journal of Science Education
Volume 26 Issue 2 February 2021 pp 257-274 General Article
In Part I , we introduced the idea of a Lax pair and ex-plained how it could be used to obtain conserved quantities for systems of particles. Here, we extend these ideas to continuum mechanical systems of ﬁelds such as the linear wave equation for vibrations of a stretched string and the Kortewegde Vries (KdV) equation for water waves. Unlike the Lax matrices for systems of particles, here Lax pairs are diﬀerential operators. A key idea is to view the Lax equation as a compatibility condition between a pair of linear equations. This is used to obtain a geometric reformulation of the Lax equation as the condition for a certain curvature to vanish. This ‘zero curvature representation’ then leads to a recipe for ﬁnding (typically an inﬁnite sequence of) conserved quantities.
Volume 26 | Issue 10