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Maximal intensity higher-order Akhmediev breathers of the nonlinear Schrodinger equation and their systematic generation

It is well known that Akhmediev breathers of the nonlinear cubic Schrodinger equation can be superposed nonlinearly via the Darboux transformation to yield breathers of higher order. Surprisingly, we find that the peak height of each Akhmediev breather only adds {\it linearly} to form the peak height of the final breather. Using this new peak-height formula, we show that at any given periodicity, there exist a unique high-order breather of maximal intensity. Moreover, these high-order breathers form a continuous hierarchy, growing in intensity with increasing periodicity. For any such higher-order breather, a simple initial wave function can be extracted from the Darboux transformation to dynamically generate that breather from the nonlinear Schrodinger equation.