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In this work, the Riemann-Hilbert problem for the 3-component Manakov systemis formulated on the basis of the corresponding $4\times 4$ matrix spectralproblem. Furthermore, by applying the nonlinear steepest descent techniques toan associated $4\times 4$ matrix valued Riemann-Hilbert problem, we can findleading-order asymptotics for the Cauchy problem of the 3-component Manakovsystem.

Long-time behavior for the Cauchy problem of the 3-component Manakov system