J. Chen and D.E. Pelinovsky
Periodic waves in the discrete MKDV equation: modulational instability and rogue waves
Physica D 445 (2023) 133652 (16 pages)
Abstract:
We derive the traveling periodic waves of the discrete modified Korteweg-de Vries
equation by using a nonlinearization method associated with a single eigenvalue. Modulational
stability of the traveling periodic waves is studied from the squared eigenfunction relation and
the Lax spectrum. We use numerical approximations to show that, similar to the continuous
counterpart, the family of dnoidal solutions is modulationally stable and the family of cnoidal
solutions is modulationally unstable. Consequently, algebraic solitons propagate on the dnoidal
wave background and rogue waves (spatially and temporally localized events) are dynamically
generated on the cnoidal wave background.
Keywords:
discrete modified Korteweg-de Vries equation, periodic standing waves,
modulational instability, algebraic solitons, rogue waves.