A New Kind of Bäcklund Transformation for Partial Differential Equations

Abstract

The existence of auto-Backlund transformations is generally understood as being a characteristic feature of integrable equations. The nature of such transformations changes depending on whether the equation under consideration is a partial differential equation (PDE) or an ordinary differential equation (ODE). We show in this paper how certain properties of completely integrable systems such as Hamiltonian structures and Miura maps can be used in order to derive ODE-type auto-Backlund transformations for a certain class of PDEs. We apply this method to two integrable equations which are nonisospectral extensions of the integrated modified inverse Korteweg–de Vries equation and of the integrated modified inverse Broer–Kaup system, and show that ODE-type auto-Backlund transformations exist for both of them. These auto-Backlund transformations involve shifts on the functions appearing as coefficients in the equations and they mimic the auto-Backlund transformations for the second and fourth Painleve equations. We believe these ODE-type auto-Backund transformations for PDEs to be new. Our derivation is in fact valid for a wide class of equations that includes both integrable and nonintegrable ODEs and PDEs.