Geometry of Darboux-Manakov-Zakharov systems and its application
Abstract
The intrinsic geometric properties of generalized Darboux-Manakov-Zakharov systems of semilinear partial differential equations GDMZabstract ∂2 u∂ xi∂ xj=fij(xk,u,∂ u∂ xl), 1≤ i<j≤ n, k,l∈\1,...,n\ for a real-valued function u(x1,...,xn) are studied with particular reference to the linear systems in this equation class. System (GDMZabstract) will not generally be involutive in the sense of Cartan: its coefficients will be constrained by complicated nonlinear integrability conditions. We derive geometric tools for explicitly constructing involutive systems of the form (GDMZabstract), essentially solving the integrability conditions. Specializing to the linear case provides us with a novel way of viewing and solving the multi-dimensional n-wave resonant interaction system and its modified version as well as constructing new examples of semi-Hamiltonian systems of hydrodynamic type. The general theory is illustrated by a study of these applications.