Long‐Wave Limit and Solitary Waves of the Two‐Dimensional Vector Fermi–Pasta–Ulam–Tsingou System

We investigate the continuous limit problem of the two‐dimensional vector Fermi–Pasta–Ulam–Tsingou (FPUT) lattice with diagonal and nearest‐neighbor interactions. In the continuous limit of the lattice, we consider the effect of slow transverse modulation and obtain new results without assuming the compatibility condition used previously in the literature. When the scale of the transverse modulation effect is , we find that the long‐wave limit for modulated waves in the two‐dimensional vector lattice is the coupled Kadomtsev–Petviashvili (CKP‐II) equation. Meanwhile, when the scale of the transverse modulation effect is and the energy coefficient , we prove that the long‐wave limit for modulated waves in the two‐dimensional vector lattice is the coupled Korteweg‐de Vries (CKdV) equation. This result complements Herrmann's work, which only considered the case of a nonzero energy coefficient. In our results, the direction of wave propagation is unrestricted, and the influence of diagonal interactions on the continuous limit problem of the lattice is elucidated.