# Method for solving hyperbolic systems with initial data on non-characteristic manifolds with applications to the shallow water wave equations

## TL;DR

This paper introduces a new method for transforming initial data on non-characteristic manifolds for hyperbolic PDEs, with applications to modeling long wave run-up in fluid mechanics, particularly in inclined bays.

## Contribution

The paper presents a novel method for converting initial conditions on non-characteristic manifolds into standard initial conditions for hyperbolic systems, enabling new solutions in fluid mechanics.

## Key findings

- Effective transformation method for hyperbolic PDE initial data
- Complete solution for long wave run-up in inclined bays
- Application to fluid mechanics problems involving nonzero initial velocity

## Abstract

We are concerned with hyperbolic systems of order-one linear PDEs originated on non-characteristic manifolds. We put forward a simple but effective method of transforming such initial conditions to standard initial conditions (i.e. when the solution is specified at an initial moment of time). We then show how our method applies in fluid mechanics. More specifically, we present a complete solution to the problem of long waves run-up in inclined bays of arbitrary shape with nonzero initial velocity.