Pseudopotentials and integrability properties of the Burgers' equation

This chapter discusses Riccati-type pseudopotentials and their applications. This type of pseudopotential generates the Lax equations, auto-Backlund transformation, and singularity manifold equation of the corresponding nonlinear evolution (NLE) equation in 1+ 1 dimensions by using the properties of the Riccati ordinary differential equation. This technique can be generalized to NLE equations in 2+1 dimensions by imposing the defining equations of the pseudopotential to be of a Riccati-type in one-space variable. Lax equations and auto-Backlund transformations for an equation with higher-order scattering can also be derived if a pseudopotential exists such that its defining equations are of a type given by a member of the Riccati-chain. A topic related to Riccati-type pseudopotentials is the derivation of novel S-integrable equations and their auto-Backlund transformation from the singularity manifold equations and their invariance under the Mobius group, respectively. A well-known example is the link between the Harry Dym and the singularity manifold equation of the Korteweg–de Vries equation. The chapter also considers equations that are not S-integrable. These equations possess Riccati-type pseudopotentials that derive from local conservation laws.