New integrable equations of fourth order and higher degree related to Cosgrove's equation

We give a general formulation of the algorithm of Fokas and Ablowitz, which then allows us to obtain transformations for nth order ordinary differenti al equations, to equations of the same order but perhaps of higher degree. Previously this algorithm has been used to obtain transformations for the s ix second order equations defining new transcendental functions discovered by Painleve and co-workers, either to other equations in the Painleve class ification or to equations of second order and second degree. As an example of our approach we consider a new fourth order ordinary differential equati on due to Cosgrove which is believed to define a new transcendent. We obtai n transformations relating this equation to other fourth order ordinary dif ferential equations, of degrees greater than or equal to2. All of these tra nsformations, as well as the corresponding higher degree differential equat ions, all of which have the Painleve property, are new. (C) 2001 American I nstitute of Physics.