Lee-Huang-Yang (LHY) fluids are an exotic quantum matter emerged in a Bose-Bose mixture where the mean-field interactions, interspecies attraction $(g_{12})$ and intraspecies repulsive $(g_{11}, g_{22})$, are tuned to cancel completely when $g_{12}=-sqrt{g_{11}g_{22}}$ and atom number $N_2=sqrt{g_{11}/g_{22}}N_1$, and as such the fluids are purely dominated by beyond mean-field (quantum many-body) effect -- quantum fluctuations.Three-dimensional LHY fluids were proposed in 2018 and demonstrated by the same group from Denmark in recent ultracold atoms experiments [T. G. Skov,et al., Phys. Rev. Lett. 126, 230404], while their low-dimensional counterparts remain mysterious even in theory. Herein, we derive the Gross-Pitaevskii equation of one-dimensional LHY quantum fluids in two-component Bose-Einstein condensates, and reveal the formation, properties, and dynamics of matter-wave structures therein. An exact solution is found for fundamental LHY fluids. Considering a harmonic trap, approximate analytical results are obtained based on variational approximation, and higher-order nonlinear localized modes with nonzero nodes $Bbbk=1$ and $2$ are constructed numerically. Stability regions of all the LHY nonlinear localized modes are identified by linear-stability analysis and direct perturbed numerical simulations. Movements and oscillations of single localized mode, and collisions between two modes, under the influence of different incident momenta are also studied in dynamical evolutions. The predicted results are available to quantum-gas experiments, providing a new insight into LHY physics in low-dimensional settings.
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