A New Kind of Bäcklund Transformation for Partial Differential Equations

The existence of auto-Bäcklund 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-Bäcklund 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-Bäcklund transformations exist for both of them. These auto-Bäcklund transformations involve shifts on the functions appearing as coefficients in the equations and they mimic the auto-Bäcklund transformations for the second and fourth Painlevé equations. We believe these ODE-type auto-Bäckund 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.