<p>This paper investigates families of novel exact soliton solutions in the form of hyperbolic, rational, exponential, trigonometric functions and their combinations for a continuous model of cold bosonic atoms in a zig-zag optical lattice via modified generalised exponential rational function method. The continuous model has been derived from the discrete model using the continuum approximation. The 3D, 2D and density graphs of the amplitude profile of the periodic, dark and bright singular solitons are plotted for analysing the effect of first-nearest-neighbour (FNN) hopping, second-nearest-neighbour (SNN) hopping, the strength of the boson–boson interaction, boson number and group velocity dispersion coefficient. As the value of FNN enhances, the soliton amplitude increases, while the shapes of the amplitude profile remain preserved. Further, modulation instability (MI) in the continuous model is investigated. The MI gain is studied against the wave number, FNN hopping, SNN hopping, boson number and initial incidence power. The effect of SNN hopping is higher than the effect of FNN hopping. It is observed that the proper choice of the parameters can manage the soliton solutions and MI for the continuous model.</p>