Symmetry breaking in the nonlinear stage of modulation instability for the Kundu-Eckhaus equation: Nonlinear excitations and interactions with solitons

In the context of the Kundu-Eckhaus equation, we study the nonlinear stage of modulational instability (MI) driven by a purely continuous spectrum and then analyze the nonlinear interactions in the presence of an additional soliton generated by the discrete spectrum. Specifically, investigating the characteristics of spontaneous oscillations generated by nonlinear MI, we show that the oscillation structure asymptotically evolves into a soliton ensemble as time goes to infinity. Meanwhile, the quintic and derivative nonlinearities can modulate the boundary expansion velocities and peak velocities of the oscillation structure, leading to the loss of spatial symmetry. When the presence of a discrete spectrum gives rise to an extra soliton independent of the initial perturbation, we identify four types of interactions between the soliton and the oscillation structure and accordingly present the partition of the discrete spectrum on the spectral plane, which varies with the strength of the higher-order nonlinearities.