Continuous limit, higher-order rational solutions and relevant dynamical analysis for Belov–Chaltikian lattice equation with 3\(\times \)3 Lax pair

Belov–Chaltikian (BC) lattice equation, which is related to the research of lattice analogues of W-algebras, is under consideration in this work. This equation may be viewed as an extension of the Volterra lattice equation. Firstly, we correspond BC lattice equation to several continuous equations under the continuous limit. Secondly, based on the known \(3\times 3\) matrix form Lax pair of this discrete equation, we construct its discrete generalised \((m, 3N-m)\)-fold Darboux transformation for the first time and successfully popularise this technique from \(2\times 2\) Lax pair to \(3\times 3\) Lax pair. Finally, by applying the resulting Darboux transformation, we get its higher-order rational solutions and analyse their singular trajectories and dynamics using the graphics and limit-state analysis.