On the integrability of a class of Monge-Ampere equations

Zheltukhın, Kostyantyn
We give the Lax representations for the elliptic, hyperbolic and homogeneous second order Monge-Ampere 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-Ampere equations. Local as well nonlocal conserved densities are obtained.


Gardner's deformations of the Boussinesq equations
Karasu, Atalay (IOP Publishing, 2006-09-15)
Using the algebraic method of Gardner's deformations for completely integrable systems, we construct recurrence relations for densities of the Hamiltonians for the Boussinesq and the Kaup-Boussinesq equations. By extending the Magri schemes for these equations, we obtain new integrable systems adjoint with respect to the initial ones and describe their Hamiltonian structures and symmetry properties.
Time-dependent recursion operators and symmetries
Gurses, M; Karasu, Atalay; Turhan, R (Informa UK Limited, 2002-05-01)
The recursion operators and symmetries of nonautonomous, (1 + 1) dimensional integrable evolution equations are considered. It has been previously observed hat he symmetries of he integrable evolution equations obtained through heir recursion operators do not satisfy the symmetry equations. There have been several attempts to resolve his problem. It is shown that in the case of time-dependent evolution equations or time-dependent recursion operators associativity is lost. Due to this fact such recursion ope...
Finite action Yang-Mills solutions on the group manifold
Dereli, T; Schray, J; Tucker, RW (IOP Publishing, 1996-08-21)
We demonstrate that the left (and right) invariant Maurer-Cartan forms for any semi-simple Lie group enable solutions of the Yang-Mills equations to be constructed on the group manifold equipped with the natural Cartan-Killing metric. For the unitary unimodular groups the Yang-Mills action integral is finite for such solutions. This is explicitly exhibited for the case of SU(3).
GÜRSES, METİN; Karasu, Atalay; TURHAN, REFİK (Informa UK Limited, 2010-03-01)
We construct the recursion operators for the non-commutative Burgers equations using their Lax operators. We investigate the existence of any integrable mixed version of left- and right-handed Burgers equations on higher symmetry grounds.
Integrable boundary value problems for elliptic type Toda lattice in a disk
Guerses, Metin; Habibullin, Ismagil; Zheltukhın, Kostyantyn (AIP Publishing, 2007-10-01)
The concept of integrable boundary value problems for soliton equations on R and R+ is extended to regions enclosed by smooth curves. Classes of integrable boundary conditions in a disk for the Toda lattice and its reductions are found. (C) 2007 American Institute of Physics.
Citation Formats
J. C. BRUNELLI, M. GÜRSES, and K. Zheltukhın, “On the integrability of a class of Monge-Ampere equations,” REVIEWS IN MATHEMATICAL PHYSICS, pp. 529–543, 2001, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/46517.