# On the unique continuation property of solutions of the three-dimensional Zakharov-Kuznetsov equation

## TL;DR

This paper establishes a unique continuation property for solutions of the three-dimensional Zakharov-Kuznetsov equation, showing that solutions with sufficiently fast decay at two different times must be identical.

## Contribution

It proves a new unique continuation result for the 3D Zakharov-Kuznetsov equation based on decay conditions at two time points.

## Key findings

- Solutions with exponential decay at two times are identical.
- The decay condition ensures the solutions coincide.
- The result applies to sufficiently smooth solutions.

## Abstract

We prove that if the difference of two sufficiently smooth solutions of the three-dimensional Zakharov-Kuznetsov equation $$\partial_{t}u+\partial_{x}\triangle u+u\partial_{x}u=0 \text{,}\quad (x,y,z)\in\mathbb R^3, \;t\in[0,1],$$ decays as $e^{-a(x^2+y^2+z^2)^{3/4}}$ at two different times, for some $a>0$ large enough, then both solutions coincide.