On a class of singular nonlinear first order partial differential equations

TL;DR

This paper studies a specific class of singular nonlinear first order PDEs with complex variables, providing conditions under which all solutions near the origin can be explicitly determined.

Contribution

It characterizes solutions of a class of singular nonlinear PDEs with complex variables, extending understanding of their local behavior near the origin.

Findings

Explicit solutions near the origin are determined under certain conditions.

The class of PDEs considered includes singular nonlinear equations with complex variables.

Conditions for the existence and uniqueness of solutions are established.

Abstract

In this paper, we consider a class of singular nonlinear first order partial differential equations $t(\partial u/\partial t)=F(t,x,u, \partial u/\partial x)$ with $(t,x) \in \mathbb{R} \times \mathbb{C}$ under the assumption that $F(t,x,z_1,z_2)$ is a function which is continuous in $t$ and holomorphic in the other variables. Under suitable conditions, we determine all the solutions of this equation in a neighborhood of the origin.