# Method of Characteristics in Smooth Problems

• Arik Melikyan

## Abstract

Consider general nonlinear first order partial differential equation (PDE):
$$F\left({x,u\left( x \right),p\left( x \right)} \right) = 0,x \in D \subset {\mathbb{R}^n}\left( {p = {{\partial u} \mathord{\left/ {\vphantom {{\partial u} {\partial x = {u_x}}}} \right. \kern-\nulldelimiterspace} {\partial x = {u_x}}}} \right)$$
(1.1)
Here x = (x 1 , x n ) is n-dimensional vector of the space $${\mathbb{R}^n}$$, D is an open neighborhood of a reference point x* $${\mathbb{R}^n}$$ u is the scalar unknown function, u: D→ $${\mathbb{R}^n}$$ 1 , and p = (P1,, p n ) is the vector of its gradient, pi =$$\partial$$u/$$\partial$$x i , i = 1,…, n. The scalar function F will be called the Hamiltonian, F: N→ $${\mathbb{R}^1}$$ , where $$N = D \times {\mathbb{R}^1} \times {\mathbb{R}^n}$$ is a domain in (2n + 1)-dimensional space of (x, u, p) $$\in {\mathbb{R}^{2n + 1}}$$

## Keywords

Characteristic Vector Cauchy Problem Characteristic Point Characteristic System Implicit Function Theorem
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