Novel soliton breathers for the higher-order Ablowitz-Kaup-Newell-Segur hierarchy

In the framework of the nonautonomous exactly integrable higher-order equations belonging to the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy, the method for systematic construction of the bound soliton states (localized breathers on the zero background) is presented, and general and specific conditions for their appearance are formulated. We show that both all parameters of solitons forming the breather, and the varying dispersion and nonlinearities should be appropriately chosen in accordance with the exact integrability conditions of the models. Novel breather solutions of the higher-order nonlinear evolution equations with vanishing boundary conditions generally move with varying amplitudes and velocities adapted to variations of the dispersion, nonlinearity, and gain or losses. We demonstrate that novel soliton breather solutions substantially extend the breather concept known so far. The revealed specific conditions for the breather formation can be generalized and applied to the evolution equations of the next, more and more higher orders of the AKNS hierarchy.