Symmetries, exact solutions, and nonlinear superposition formulas for two integrable partial differential equations (Q2738075)

The author constructs exact solutions for two sixth-order \(1+1\)-dimensional nonlinear equations recently introduced by herself and \textit{A. Pickering} [Bäcklund transformations for two new integrable partial differential equations, Europhys. Lett. 47, No. 1, 21-24 (1999)]. Both equations arise as reductions of certain 2+1 integrable systems, have a third-order nonisospectral scattering problem, and have among their reductions the suspected new Painlevé transcendent recently introduced by Cosgrove. Found are solutions that arise from similarity reductions with respect to translational and scaling symmetries as well as multisoliton solutions obtained by the Darboux transformation and the accompanying nonlinear superposition principle.