Modulational instability and droplet formation in Bose-Bose mixtures with Lee-Huang-Yang correction and polaron-like impurity

This letter studies the excitation of nonlinear patterns on the one-dimensional (1D) quantum droplet in the presence of polaron-like impurity under modulational instability (MI) activation. Analytically, it is shown that strong coupling between the impurity and Bose species expands the instability domain, especially in the quantum droplet regime, while pronounced self-repulsive nonlinearity tends to restrict the MI to regions of strong interspecies interactions. In the process, the wavenumber spectrum, capable of supporting MI, also shrinks. Direct numerical simulations confirm the existence of coupled boson-impurity wave patterns, mainly governed by trains of solitary waves of different profiles for the impurity and quasi-droplet waveforms for the boson. Additionally, the two components are mutually trapped, which enhances energy transfer between them, leading to the formation of quasi-droplet structures under a suitable balance between the back action of the impurity on the boson, the Lee-Huang-Yang term, as well as the intrinsic self-repulsive nonlinearity.