Recursion operators for a class of integrable third-order evolution equations

Abstract

We consider $u_t=u^{\alpha }{u}_{xxx}+n(u)u_x{u}_{xx}+m(u)u_x^3+r(u)u_{xx}+p(u)u_x^2+q(u)u_x+s(u)$ with $\alpha =0$ and $\alpha =3$, for those functional forms of $m,n,p,q,r,s$ for which the equation is integrable in the sense of an infinite number of Lie-Bäcklund symmetries. Local $x$- and $t$-independent recursion operators that generate these infinite sets of symmetries are obtained for the equations. A combination of potential forms, hodograph transformations and $x$-generalised hodograph transformations are applied to the obtained equations.