# The Replacement Rule for Nonlinear Shallow Water Waves

## TL;DR

This paper introduces a replacement rule for nonlinear PDEs describing shallow water waves, providing a simple way to relate envelope width, amplitude, and group velocity without solving the equations.

## Contribution

It proposes a nearly universal qualitative relation (replacement rule) for parameters of localized solutions in nonlinear PDEs modeling shallow water waves.

## Key findings

- The rule applies to KdV, C-H, and BBM equations.
- It offers a qualitative understanding without explicit solutions.
- The method is broadly valid for such nonlinear PDEs.

## Abstract

When a $(1+1)$-dimensional nonlinear PDE in real function $\eta(x,t)$ admits localized traveling solutions we can consider $L$ to be the average width of the envelope, $A$ the average value of the amplitude of the envelope, and $V$ the group velocity of such a solution. The replacement rule (RR or nonlinear dispersion relation) procedure is able to provide a simple qualitative relation between these three parameters, without actually solve the equation. Examples are provided from KdV, C-H and BBM equations, but the procedure appears to be almost universally valid for such $(1+1)$-dimensional nonlinear PDE and their localized traveling solutions \cite{3}.