# Solitons in the Korteweg–De Vries & Zakharov–Kuznetsov equations

Here is an implementation of the Korteweg–De Vries (KdV) equation given by

\[\pd{\phi}{t}=-\pdn{\phi}{x}{3}-6\phi \pd{\phi}{x},\]which is a very simple model of solitons, as described at the bottom of the page for the nonlinear Schrödinger equation.

The interactive simulation starts with two solitons of different amplitudes and speeds, with the larger one moving more quickly and hence overtaking the smaller one at first. While they occupy the same space, these solitons cannot be clearly distinguished as their ampltiudes locally add together, but as the faster soliton moves more quickly it eventually separates from the slower soliton, and neither speed nor amplitude of either soliton is changed.

This example was helpfully constructed by Paul Sutcliffe.

## 2D Vortical Solitons

There are also 2D analogues of solitons sometimes called vortical solitons. The modified Zakharov–Kuznetsov model is given by

\[\pd{u}{t} = -\frac{\partial^3 u}{\partial x^3}-\frac{\partial^3 u}{\partial x \partial y^2} - u \pd{u}{x}-b\nabla^4 u,\]where $b$ is a small dissipative term used to reduce radiation. This simulation shows one such vortical soliton moving in the positive $x$ direction.