Long-Time Asymptotics for the Spin-1 Gross–Pitaevskii Equation

Abstract

On the basis of the spectral analysis of the $$4\times 4$$ Lax pair associated with the spin-1 Gross–Pitaevskii equation and the scattering matrix, the solution to the Cauchy problem of the spin-1 Gross–Pitaevskii equation is transformed into the solution to the corresponding Riemann–Hilbert problem. The Deift–Zhou nonlinear steepest descent method is extended to the Riemann–Hilbert problem, from which a model Riemann–Hilbert problem is established with the help of distinct factorizations of the jump matrix for the Riemann–Hilbert problem and a decomposition of the matrix-valued spectral function. Finally, the long-time asymptotics of the solution to the Cauchy problem of the spin-1 Gross–Pitaevskii equation is obtained.