Generic transverse stability of kink structures in atomic and optical nonlinear media with competing attractive and repulsive interactions

We demonstrate the existence and stability of one-dimensional (1D) topological kink configurations immersed in higher-dimensional bosonic gases and nonlinear optical setups. Our analysis pertains, in particular, to the two- and three-dimensional extended Gross-Pitaevskii models with quantum fluctuations describing droplet-bearing environments but also to the two-dimensional cubic-quintic nonlinear Schr\"odinger equation containing higher-order corrections to the nonlinear refractive index. Contrary to the generic dark soliton transverse instability, the kink structures are generically robust under the interplay of low-amplitude attractive and high-amplitude repulsive interactions. A quasi-1D effective potential picture dictates the existence of these defects, while their stability is obtained numerically and analytically through linearization analysis and direct dynamics in the presence of external fluctuations showcasing their unprecedented resilience. These generic (across different models) findings should be detectable in current cold atom and optics experiments, offering insights towards controlling topological excitations.