# Geometry of Darboux-Manakov-Zakharov systems and its application - Mathematics > Differential Geometry

Geometry of Darboux-Manakov-Zakharov systems and its application - Mathematics > Differential Geometry - Download this document for free, or read online. Document in PDF available to download.

Abstract: The intrinsic geometric properties of generalized Darboux-Manakov-Zakharovsystems of semilinear partial differential equations \label{GDMZabstract}\frac{\partial^2 u}{\partial x i\partial x j}=f {ij}\Bigx k,u,\frac{\partialu}{\partial x l}\Big, 1\leq i

.,n\} for a real-valuedfunction $ux 1,

.,x n$ are studied with particular reference to the linearsystems in this equation class.System ef{GDMZabstract} will not generally be involutive in the sense ofCartan: its coefficients will be constrained by complicated nonlinearintegrability conditions. We derive geometric tools for explicitly constructinginvolutive systems of the form ef{GDMZabstract}, essentially solving theintegrability conditions. Specializing to the linear case provides us with anovel way of viewing and solving the multi-dimensional $n$-wave resonantinteraction system and its modified version as well as constructing newexamples of semi-Hamiltonian systems of hydrodynamic type. The general theoryis illustrated by a study of these applications.

Author: ** Peter J. Vassiliou**

Source: https://arxiv.org/