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There exists a critical speed of propagation of the line solitons in theZakharov-Kuznetsov (ZK) equation such that small transversely periodicperturbations are unstable for line solitons with larger-than-critical speedsand orbitally stable for those with smaller-than-critical speeds. The normalform for transverse instability of the line soliton with a nearly criticalspeed of propagation is derived by means of symplectic projections andnear-identity transformations. Justification of this normal form is providedwith the energy method. The normal form predicts a transformation of theunstable line solitons with larger-than-critical speeds to the orbitally stabletransversely modulated solitary waves.

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