Do N-soliton breathers exist for the Hirota equation models?

Abstract

We reveal the method for systematic construction of unexpected bound soliton states (localized breathers on the zero background) and formulate specific conditions for their appearance both in the framework of the canonical and the generalized nonautonomous Hirota models. These conditions show that both all parameters of solitons forming the breather and the dispersion and nonlinearity variations cannot be chosen independently, they are related by the exact integrability of the model. Novel N-soliton breathers derived for the nonautonomous Hirota equation with vanishing boundary conditions differ in that they generally move with varying amplitudes and velocities adapted to variations of the dispersion, nonlinearity, and gain, or losses. We apply the specific conditions for the breather formation to derive novel soliton breather solutions of the nonlinear Schrödinger and the complex modified Korteweg–de Vries equations. Consequently, we demonstrate that the way is opened to generalize and apply the presented method for other evolution equations of the Ablowitz–Kaup–Newell–Segur hierarchy.