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

In the context of the line solitons in the Zakharov–Kuznetsov (ZK) equation, there exists a critical speed of propagation such that small transversely periodic perturbations are unstable if the soliton speed is larger than the critical speed and orbitally stable if the soliton speed is smaller than the critical speed. 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.