Symmetries, exact solutions, and nonlinear superposition formulas for two integrable partial differential equations

Abstract

We recently introduced two new sixth-order partial differential equations (PDEs) associated with third-order scattering problems. Here we extend our study of these PDEs by considering the construction of exact solutions both by using the method of symmetry reduction due to Lie, and by using their Darboux transformations (DTs). Amongst the ordinary differential equations (ODEs) obtained by symmetry reduction is an ODE due to Cosgrove that is believed to define a new Painlevé transcendent. This ODE provides soliton solutions for our integrable PDEs that include arbitrary functions of time. The DTs for our PDEs allow the recovery of these solutions and in addition provide other solutions which are not associated with Lie symmetries (either classical or nonclassical). We also consider the iteration of the corresponding Bäcklund transformations (BTs) for these PDEs. The theorem of permutability allows us to reduce this process of iterating the DT from one of solving a third-order linear equation (the spatial part of the Lax pair) to that of solving either a second-order linear equation (for one PDE), or quite remarkably to that of solving a first-order linear equation (for the other PDE). These linear differential equations have coefficients involving three previous solutions of the PDE, and are a natural extension of the linear algebraic equation found by applying the theorem of permutability to the Korteweg–de Vries equation.