Commutator identities on associative algebras and integrability of nonlinear pde's

It is shown that commutator identities on associative algebras generate solutions of linearized integrable equations. Next, a special kind of the dressing procedure is suggested that in a special class of integral operators enables to associate to such commutator identity both nonlinear equation and its Lax pair. Thus problem of construction of new integrable pde's reduces to construction of commutator identities on associative algebras.Comment: 12 page

Towards an Inverse Scattering theory for non decaying potentials of the heat equation

The resolvent approach is applied to the spectral analysis of the heat equation with non decaying potentials. The special case of potentials with spectral data obtained by a rational similarity transformation of the spectral data of a generic decaying potential is considered. It is shown that these potentials describe $N$ solitons superimposed by Backlund transformations to a generic background. Dressing operators and Jost solutions are constructed by solving a DBAR-problem explicitly in terms of the corresponding objects associated to the original potential. Regularity conditions of the potential in the cases N=1 and N=2 are investigated in details. The singularities of the resolvent for the case N=1 are studied, opening the way to a correct definition of the spectral data for a generically perturbed soliton.Comment: 22 pages, submitted to Inverse Problem

The formation of shear-band/fracture networks from a constitutive instability: Theory and numerical experiment

Journal of Geophysical Research, Solid Earth, v. 112, p. B11404, 2007. http://dx.doi.org/10.1029/2007JB005026International audienc