Explicit solution for the hyperbolic homogeneous scalar one-dimensional conservation law

TL;DR

The paper presents an explicit integral formula for solving the initial value problem of nonlinear scalar hyperbolic conservation laws in one dimension, applicable for analytic initial data and flux functions.

Contribution

It introduces a new explicit integral solution formula for scalar conservation laws with analytic initial conditions and flux functions.

Findings

Valid for as long as the solution remains analytic

Applicable to any flux function and initial condition that are analytic

Provides a direct method to solve the initial value problem

Abstract

A complex integral formula provides an explicit solution of the initial value problem for the nonlinear scala 1D equation $u_t+[f(u)]_x = 0$, for any flux $f(u)$ and initial condition $u_0(x)$ that are analytic. This formula is valid at least as long as $u$ remains analytic.