Nonlinear Schrodinger equation is a completely integrable Hamiltonian system. It admits infinite number of conserved quantities. It also admits rational solution (or rogue wave solution).The main character of this solution is unstable and it appears nowhere without a trace. So my question is what happen to the energy of the rogue wave when it goes off.
I am not sure what is meant by "goes off". If I assume this refers to the occurrence, the key to the answer you already stated: conserved quantities of the the Hamiltonian system. In the absence of currents, energy is conserved. If the "rogue wave" is due to modulation instability (for example see Yuen, H. C. and B. M. Lake (1980), Instabilities of waves on deep water, Annu. Rev. Fluid Mech., 12(1), 303-334.; Yuen, H. C. and B. M. Lake (1982), Nonlinear dynamics of deep-water gravity waves, Adv. Appl. Mech., 22(1), 67.) then groups may from monochromatic waves. After a while, the groups disperse and the monochromatic wave is recovered. In this sense, the rogue wave appears and disappears on a time scale related to the steepness. This is an analog to Fermi-Pasta-Ulam recurrence.
I am a believer in the use of NLSE for optical phenomena not for surface waves. There is some work on the importance of transverse shear flow in generating these. When you say "goes off" do you mean that the wave is suddenly dissipating? I did a little calculation recently that gives the energy of such a bound wave group as borrowed from the surface shear energy so the vanishing of the shear allows the wave to spread into the spectral components and simply become lower and wider.
Article Surface Shear, Persistent Wave Groups and Rogue Waves
For NLS rogue wave is a rational solution asymptotic to a positive number as x goes to infinity, while the usual envelope soliton solution is asymptotic to zero. When you check conservation laws asymptotic condition should be consisdered.
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