Solutions with separated variables and breather structures in the (1+1)-dimensional nonlinear systems

In this Letter, the multilinear variable separation approach is extended to (1+1)-dimensional nonlinear systems, and the independent variables of these systems are totally separated. A common formula with some arbitrary functions is derived to describe suitable physical quantities for some (1+1)-dimensional models such as the negative KdV hierarchy, the long-wave-short-wave resonant interaction equation, the Ito system, the shallow water wave equations and the coupled integrable dispersionless equations. Based on the formula and by selecting appropriate functions, rich breather structures, such as soliton-type, peakon-type, compacton-type, foldon-type breathers, can be investigated.