Long‐Time Asymptotics of Solutions for Any n$n$‐Component Focusing Nonlinear Schrödinger Equations in Spacetime Solitonic Regions

ABSTRACT The study of any ‐order () matrix Riemann–Hilbert problem with poles related to nonlinear integrable systems is still an open issue. In this paper, we present a new method to construct the initial ‐order matrix Riemann–Hilbert problem and its residue conditions related to any ‐component focusing nonlinear Schrödinger (NLS) equations via the ‐Lax pair and its adjoint form. By extending the ‐steepest descent method to this matrix Riemann–Hilbert problem, we analyze the Cauchy problem of the ‐component focusing NLS equations. With the aid of distinct factorizations of the ‐order jump matrix for the Riemann–Hilbert problem and a decomposition of the matrix‐valued spectral function, a model Riemann–Hilbert problem is established. Finally, we find the long‐time asymptotics for the Cauchy problem of the ‐component focusing NLS equations in spacetime solitonic regions. The idea can also be extended to other ‐component nonlinear integrable systems.