Solutions of Analytical Systems of Partial Differential Equations

TL;DR

The paper studies analytic systems of partial differential equations, both linear and nonlinear, and derives integrability conditions that ensure local solvability. It shows that the vanishing of curvature-type tensors $R_{tsuv}$ (and their multi-index nonlinear analogues $R_{\\alpha\\beta uv}$) is necessary and sufficient for the existence of solutions in a neighborhood of the origin. Solutions are constructed as convergent functional-series in the variables $x=(x_1,\dots,x_k)$ with prescribed initial data $y_s(0,\dots,0)=C_s$. For the linear homogeneous case, Theorem 2.1 provides a unique local solution of the form $y_r = \sum_{s=1}^n \sum_{w_1,\dots,w_k \ge 0} \frac{(-x_1)^{w_1}}{w_1!} \cdots \frac{(-x_k)^{w_k}}{w_k!} P^{<w_1,\dots,w_k>}_{rs} C_s$, with $P^{<0,\dots,0>}_{ts} = \delta_{ts}$ and a recursive relation for $P^{<w>}$; for the nonlinear case, Theorem 3.2 extends the construction via Laurent expansions in the unknowns to obtain a similar convergent series with coefficients $P^{<w_1,...,w_k>}_{i_1\dots i_n j_1\dots j_n}$ and initial constants $C^{j}$, yielding a unique local tensorial solution. The results relate integrability to curvature in differential geometry and offer a systematic, algebraic way to obtain tensorial local solutions, with potential applications to nonlinear connections and Frenet-type equations.

Abstract

In this paper are examined general classes of linear and non-linear analytical systems of partial differential equations. Indeed the integrability conditions are found and if they are satisfied, the solutions are given as functional series in a neighborhood of a given point (x=0).