Decomposition of Higher-Order Equations of Monge–Ampère Type

Abstract

It is demonstrated that the fourth-order PDE

$$\det \left( {\begin{array}{*{20}c} {f_{xxxx} } & {f_{xxxt} } & {f_{xxtt} } \\ {f_{xxxt} } & {f_{xxtt} } & {f_{xttt} } \\ {f_{xxtt} } & {f_{xttt} } & {f_{tttt} } \\ \end{array} } \right) = 0$$

decouples in a pair of identical second-order Monge–Ampère equations,

$$u_{xx} u_{tt} - u_{xt}^2 = 0{ and \upsilon }_{xx} {\upsilon }_{tt} - {\upsilon }_{xt}^2 = 0$$

by virtue of the Bäcklund-type relation

$$u_{xx} u_{tt} - u_{xt}^2 = 0{and \upsilon }_{xx} {\upsilon }_{tt} - {\upsilon }_{xt}^2 = 0$$

A higher order generalization of this decomposition ia also proposed.