Rogue waves in the sea: observations, physics, and mathematics

Rogue waves arising on the standing periodic waves in the Ablowitz–Ladik equation

“…The red dots on the 𝜆-plane show zeros {±𝜆 1 , ±𝜆 −1 1 , ±𝜆 2 , ±𝜆 −1 2 } of the polynomial 𝑃(𝜆) in ( 29) with 𝜔 = dn(𝛼; 𝑘) cn 2 (𝛼; 𝑘) and 𝐹 1 = (1 − 𝑘 2 ) sn 4 (𝛼; 𝑘) cn 4 (𝛼; 𝑘) . Applying the ordering 0 < 𝜆 1 < 𝜆 2 < 1, the two roots 𝜆 1 and 𝜆 2 were obtained in Ref.…”

“…Lemma 7. Let {𝑢 𝑛 (𝑡)} 𝑛∈ℤ be a solution of the AL equation (1), {(𝑝 𝑛 (𝑡), 𝑞 𝑛 (𝑡)) T } 𝑛∈ℤ be a nontrivial solution to the linear system (4) with 𝜆 = 𝜆 1 , and {𝜑 𝑛 (𝑡)} 𝑛∈ℤ be any solution to the linear system (4) with arbitrary 𝜆 ∈ ℂ. Then, { û𝑛 (𝑡)} 𝑛∈ℤ given by…”

“…1,2 Complicated wave patterns can be expressed analytically by using exact solutions of the NLS equation for periodic and double-periodic standing waves (see review in Refs. 3,4). The standing periodic waves are known to be modulationally unstable 5,6 and rogue waves (localized perturbations in space and time) have been observed on their backgrounds in numerical experiments.…”