Classification of Integrable Evolution Equations of the Form $u_t=u_{xxx}+f(t,x,u,u_x,u_{xx})$

Abstract

We obtain the classification of integrable equations of the form $u_t=u_{xxx}+f(t,x,u,u_x,u_{xx})$ using the formal symmetry method of Mikhailov et al [A.V.Mikhailov, A.B.Shabat and V.V.Sokolov, in {\it What is Integrability} edited by V.E. Zakharov (Springer-Verlag, Berlin 1991)]. We show that all such equations can be transformed to an integrable equation of the form $v_t=v_{xxx}+f(v,v_x,v_{xx})$ using transformations $\Phi(x,t,u,v,u_x,v_x)=0$, and the $u_{xx}$ dependence can be eliminated except for two equations.