Multi-component Nonlinear Schrödinger Equations with Nonzero Boundary Conditions: Higher-Order Vector Peregrine Solitons and Asymptotic Estimates

The any multi-component nonlinear Schrodinger (alias n-NLS) equations with nonzero boundary conditions are studied. We first find the fundamental and higher-order vector Peregrine solitons (alias rational rogue waves (RWs)) for the n-NLS equations by using the loop group theory, an explicit $$\left( n+1\right) $$ -multiple root of a characteristic polynomial of degree $$(n+1)$$ related to the Benjamin–Feir instability, and inverse functions. Particularly, the fundamental vector rational RWs are proved to be parity-time-reversal symmetric for some parameter constraints and classified into n cases in terms of the degree of the introduced polynomial. Moreover, a systematic approach is proposed to study the asymptotic behaviors of these vector RWs such that the decompositions of RWs are related to the so-called governing polynomials $${\mathcal {F}}_\ell (z)$$ , which pave a powerful way in the study of vector RW structures of the multi-component integrable systems. The vector RWs with maximal amplitudes can also be determined via the parameter vectors, which are interesting and useful in the study of RWs for multi-component nonlinear physical systems.