Multi-quadratic equations associated with KdV-type equations

Abstract

The potential KdV equation admits dpKdV as a full discretization. Similarly, most KdV-type equations admit multi-quadratic quad equations as algebraic condition on Bäcklund transformation. These quad equations also yield three-point discrete symmetries and ensure the commutativity of two symmetries corresponding to different lattice directions. In each case, the quad equation is either an ABS equation or a multi-quadratic equation of the Atkinson--Nieszporski class. For KdV, mKdV, potential Gardner and exponential Calogero--Degasperis, these are the equations $H{2}^{\ast }$, $A{1}^{\ast }$, $Q{1}^{\ast }$ and $A{2}^{\ast }$, respectively, and for the elliptic Calogero--Degasperis equation it is the primary model $Q{4}^{\ast }$.

arXiv comment38 pages, 3 figures