| Abstract: |
| The nonlinear Schr\``{o}dinger equation (NLSE) describes the propagation of wave packets for fluids of finite depth. NLSE incorporates second order dispersion in the space-time domain and cubic nonlinearity. Solitons (exponentially decaying) and rogue waves (algebraically localized) solutions have been established. These modes have also been observed experimentally too. Third order dispersion needs to be restored for sufficiently short waves. Exact solutions can be obtained for special parameter regimes but numerical schemes will be necessary for arbitrary hydrodynamic configurations. Remarkably discrete versions of NLSE can be formulated and are applicable to lattice dynamics and optical fibers. The objective here is to construct discrete analogue of a third order NLSE where analytical advances are possible. More precisely, discrete breathers (pulsating modes) and rogue waves are derived by the Hirota bilinear transform. In contrast to the continuous case, rogue waves for discrete equations oscillate in amplitude in the growth and decay stages. Spatially periodic breathers are demonstrated and are utilized to verify low order conservation laws. Discrete breathers are of interest to many branches in physics and an analytical example will be tremendously valuable. |
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