A class of multivariate linear partial differential equations on $ C^p $

We used a new type of characteristics to solve a class of homogeneous linear multivariate partial differential equations on $ C^p $. For $ x $ in $ R^p $ and $ n $ in $ Z^p $, set $ \partial_x^n = \prod_{j = 1}^p \left(\partial/\partial x_j\right)^{n_j} $. Given square matrices $ \left\{N_j\right\} $ and $ \left\{S_n\right\} $ in $ C^{s\times s} $, set

$ \begin{align*} Y(x) = \exp \left(\sum\limits_{j = 1}^p x_jN_j\right) \mbox{ in } C^{s\times s} \end{align*} $

and $ T_n(x) = Y(x)\ S_n\ Y(-x) $ in $ C^{s\times s} $. When $ \left\{ N_j\right\} $ commute, we show that the linear partial differential equation

$ \begin{align*} \sum\limits_{n = 0_p}^q T_{n}(x)\ \partial_x^n f(x) = 0_s \mbox{ for } f(x)\mbox{ in }C^s \end{align*} $

has solutions $ f(x) = f_n(x, \nu) $ for each admissible $ n\leq sq $ and any $ \nu $ in $ C^p $ such that $ d(\nu) = 0 $, where

$ \begin{align*} d(\nu) = {\rm det}\ D(\nu), \ D(\nu) = \sum\limits_{n = 0_p}^q S_n\ \prod\limits_{j = 1}^p \left(\nu_j I_s+N_j\right)^{n_j}. \end{align*} $

The research aims to develop a new method, based on a novel type of characteristics, for solving a broad class of multivariate homogeneous linear partial differential equations with matrix coefficients of a specific exponential-conjugate form, extending classical Cauchy characteristic techniques beyond the univariate case and providing explicit basis solutions parameterized over complex surfaces.