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J. Chen and D.E. Pelinovsky

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Rogue periodic waves of the focusing nonlinear Schrodinger equation

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**Abstract:**

Rogue waves on the periodic background are considered for the nonlinear Schrodinger
(NLS) equation in the focusing case. The two periodic wave solutions are expressed by the
Jacobian elliptic functions dn and cn. Both periodic waves are modulationally unstable with
respect to long-wave perturbations. Exact solutions for the rogue waves on the periodic
background are constructed by using the explicit expressions for the periodic eigenfunctions
of the Zakharov–Shabat spectral problem and the Darboux transformations. These exact
solutions labeled as rogue periodic waves generalize the classical rogue wave (the so-called
Peregrine’s breather). The magnification factor of the rogue periodic waves is computed as
a function of the wave amplitude (the elliptic modulus). Rogue periodic waves constructed
here are compared with the rogue wave patterns obtained numerically in recent publications.

**Keywords**:

focusing nonlinear Schrodinger equation, periodic waves, rogue waves, modulational instability.