# Normal form for transverse instability of the line soliton with a nearly critical speed of propagation

## TL;DR

This paper derives a normal form to analyze the transverse instability of line solitons in the Zakharov-Kuznetsov equation near a critical speed, predicting their transformation into stable modulated waves.

## Contribution

It introduces a novel normal form for the transverse instability of line solitons near critical speed using symplectic projections and energy methods.

## Key findings

- Normal form accurately predicts soliton behavior near critical speed.
- Unstable solitons transform into stable transversely modulated waves.
- The approach provides a rigorous justification of the normal form.

## Abstract

There exists a critical speed of propagation of the line solitons in the Zakharov-Kuznetsov (ZK) equation such that small transversely periodic perturbations are unstable for line solitons with larger-than-critical speeds and orbitally stable for those with smaller-than-critical speeds. The normal form for transverse instability of the line soliton with a nearly critical speed of propagation is derived by means of symplectic projections and near-identity transformations. Justification of this normal form is provided with the energy method. The normal form predicts a transformation of the unstable line solitons with larger-than-critical speeds to the orbitally stable transversely modulated solitary waves.