We introduce a mechanism for generating higher order rogue waves (HRWs) of the nonlinear Schrödinger(NLS) equation: the progressive fusion and fission of n degenerate breathers associated with a critical eigenvalue λ0 creates an order-n HRW. By adjusting the relative phase of the breathers in the interacting area, it is possible to obtain different types of HRWs. The value λ0 is a zero point of an eigenfunction of the Lax pair of the NLS equation and it corresponds to the limit of the period of the breather tending to infinity. By employing this mechanism we prove two conjectures regarding the total number of peaks, as well as a decomposition rule in the circular pattern of an order-n HRW.

In this paper, using the Darboux transformation, we demonstrate the generation of first order breather and higher order rogue waves from a generalized nonlinear Schrödinger equation with several higher order nonlinear effects representing femtosecond pulse propagation through nonlinear silica fibre. The same nonlinear evolution equation can also describes the soliton type nonlinear excitations in classical Heisenberg spin chain. Such solutions have a parameter γ 1 denoting the strength of the higher order effects. From the numerical plots of the rational solutions, the compression effects of the breather and rogue waves produced by γ 1 are discussed in detail.