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References Ablowitz MJ, Hammack J, Henderson D, Schober CM (2000) Modulated periodic stokes waves in deep water. Phys Rev Lett 84:887–890 Ablowitz MJ, Hammack J, Henderson D, Schober CM (2001) Long time dynamics of the modulational instability of deep water waves. Physica D 152–153:416–433 Ablowitz MJ, Segur H (1981) Solitons and the inverse scattering transform. SIAM, Philadelphia Akhmediev NN, Korneev VI, Mitskevich NV (1988) N-modulation signals in a single-mode optical waveguide under nonlinear conditions.
Meanwhile, the freak wave appears only after 15th inverse growth-rates of instability. What happens after developing of instability but before formation of freak wave? During this relatively long period of time, the state of ﬂuid surface can be characterized as quasisolitonic turbulence, which consists of randomly located quasisolitons of different amplitudes moving with different group velocities. Numerical study of interaction of envelope soliton was done in Clamond and Grue (2002). Such interaction leads to formation of wave with large amplitude.
For NLS potentials with several unstable modes, iterated B¨acklund transformations will generate their entire stable and unstable manifolds, comprised of homoclinic orbits of increasing dimension up to the dimension of the invariant manifolds. Such higher-dimensional homoclinic orbits associated with two or more UMs are also known as combination homoclinic orbits. , lowest dimensional) homoclinic orbit of the plane wave potential is given by u(x,t) = ae−2ia 2t 1 + 2 cos(px)eσn t+2iφ +ρ + Ae2σn t+4iφ +2ρ , 1 + 2 cos(px)eσ1 t+ρ + Ae2σn t+2ρ (4) where A = 1/ cos2 φ , σn = ±p 4a2 − p2 , φ =sin−1 (p/2a), and p = µn = 2π n/L < a for some integer n.