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 ${\mathrm{\partial }}_x^n=\prod _{j=1}^p{\left(\mathrm{\partial }/\mathrm{\partial }{x}_j\right)}^{n_j}$. Given square matrices $\left\{N_j\right\}$ and $\left\{S_n\right\}$ in $C^{s\times s}$, set

$$\begin{array}{r}Y(x)=\exp \left(\sum _{j=1}^p{x}_j{N}_j\right)\text{in}{C}^{s\times s}\end{array}$$

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

$$\begin{array}{r}\sum _{n=0_p}^q{T}_n(x){\mathrm{\partial }}_x^{n}f(x)=0_s\text{for}f(x)\text{in}{C}^s\end{array}$$

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

$$\begin{array}{r}d(\nu )=\mathrm{d}\mathrm{e}\mathrm{t}D(\nu ),D(\nu )=\sum _{n=0_p}^q{S}_n\prod _{j=1}^p{\left({\nu }_j{I}_s+N_j\right)}^{n_j}.\end{array}$$

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.