On the Integrability of a Class of Monge–ampère Equations

Abstract

We give the Lax representations for the elliptic, hyperbolic and homogeneous second order Monge-Ampère equations. The connection between these equations and the equations of hydrodynamical type give us a scalar dispersionless Lax representation. A matrix dispersive Lax representation follows from the correspondence between sigma models, a two parameter equation for minimal surfaces and Monge-Ampère equations. Local as well nonlocal conserved densities are obtained.On the Integrability of a Class of Monge-Ampère Equations 531

The dispersionless completely integrable heavenly type Hamiltonian flows and their differential-geometric structure

“…as | | 0   , and N   The compatibility condition for the fi rst order partial differential can be reduced to As it was demonstrated in [22]. Chaplygin hydrodynamic system (8.8) is closely related with a class of completely integrable Monge type equations, whose geometric structure was also recently analyzed in [14], using a different approach, based on the Grassmann manifold embedding properties of general differential systems defi ned on jet-submanifolds.…”

“…This geometric structure made it possible to fi nd an additional relationship between seed differential forms on the torus and describe a new related infi nite hierarchy of integrable hydrodynamic systems. These systems, as it was demonstrated in [22], are closely related with a class of completely integrable Monge type equations, whose geometric structure was also recently analyzed in [14], using a different approach, based on the Grassmann manifold embedding properties of general differential systems defi ned on jet-submanifolds. The latter poses an interesting problem of fi nding relationships between different geometric approaches to describing completely integrable dispersionless differential systems.…”

On the Integrable Chaplygin Type Hydrodynamic Systems and Their Geometric Structure

“…Section 4 describes a new infinite hierarchy of the integrable Chaplygin hydrodynamic system, which is generated by the new seed element which is connected with the one found in the Section 3. The recent research [4] has revealed that these dynamical systems have strong connections with the equations of the Monge type. The geometric properties and the corresponding geometric structures were studied within the framework of the general differential systems on jet-manifolds theory, which used the embedding properties of the Grassman manifold [5].…”

Recursion operator and dispersionless rational Lax representation

“…[9,10] and for Hamiltonian formulations of the equations admitting a dispersionless Lax representation see Refs. [11][12][13][14][15]. We construct a recursion operator for a hierarchy of symmetries (6), using a dispersionless Lax representation.…”

“…The equation corresponding to the Lax function (13) has been studied in [3]. To remove degeneracy one can take the Lax function as…”