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

We recently introduced two new sixth-order partial differential equations ( PDEs) associated with third-order scattering problems. Here we extend our s tudy 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 Da rboux transformations (DTs). Amongst the ordinary differential equations (O DEs) obtained by symmetry reduction is an ODE due to Cosgrove that is belie ved to define a new Painleve transcendent. This ODE provides soliton soluti ons for our integrable PDEs that include arbitrary functions of time. The D Ts for our PDEs allow the recovery of these solutions and in addition provi de other solutions which are not associated with Lie symmetries (either cla ssical or nonclassical). We also consider the iteration of the correspondin g Backlund transformations (BTs) for these PDEs. The theorem of permutabili ty 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 o f solving either a second-order linear equation (for one PDE), or quite rem arkably to that of solving a first-order linear equation (for the other PDE ). These linear differential equations have coefficients involving three pr evious solutions of the PDE, and are a natural extension of the linear alge braic equation found by applying the theorem of permutability to the Kortew eg-de Vries equation. (C) 2000 American Institute of Physics. [S0022-2488(0 0)04407-8].