Stability of elliptic solutions for the Hirota equation

This paper proves that the elliptic solution of the Hirota equation is orbitally stable for the subharmonic perturbations: perturbations are periodic with period equal to an integer multiple of the period of the underlying solution. We do this by explicitly calculating the spectrum related to linear stability and the corresponding squared eigenfunction. Since the Hirota equation belongs to the NLS hierarchy, we introduce different members in this hierarchy. Combining this with the spectral stability results allows the construction of a suitable Lyapunov functional. Using an appropriate Krein signature, we can conclude the orbital stability of elliptic solution. The result we obtained is independent of the amplitude of the solution.