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

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