Novel conditions for soliton breathers of the complex modified Korteweg-de Vries equation with variable coefficients

We reveal novel unexpected conditions for soliton breathers of the generalized complex modified Korteweg-de Vries equation with variable coefficients and the loss (or gain) term (vc cmKdV). Novel relations between spectral parameters of two solitons giving rise to the breather substantially extend the well-known constraints for canonical soliton breathers. This finding allows us to systematically construct the variety of soliton breather solutions on a zero background of the considered model. These generalized breathers move with varying amplitudes and velocities adapted to variations of the dispersion, nonlinearity, and gain or losses. Among other things, we establish that both the standing generalized breathers and envelope breathers exist as well.