Method for Solving the Multidimensional n-Wave Resonant Equations and Geometry of Generalized Darboux-Manakov-Zakharov Systems

摘要

The intrinsic geometric properties of generalized Darboux-Manakov-Zakharov systems of semilinear partial differential equations 1 for a real-valued function u(x_1, ..., x_n) are studied with particular reference to the linear systems in this equation class. System (1) is overdetermined and will not generally be involutive in the sense of Cartan: its coefficients will be constrained by complicated nonlinear integrability conditions. We derive tools for explicitly constructing involutive systems of the form (1), essentially solving the integrability conditions. Specializing to the linear case provides us with a novel way of viewing and solving the multidimensional n-wave resonant interaction system and its modified version. For each integer n≥ 3 and nonnegative integer k, our procedure constructs solutions of the n-wave resonant interaction system depending on at least k arbitrary functions each of one variable. The construction of these solutions relies only on differentiation, linear algebra, and the solution of ordinary differential equations.