Long-time asymptotics of the focusing Kundu-Eckhaus equation with nonzero boundary conditions

The long-time asymptotics of the focusing Kundu-Eckhaus equation with nonzero boundary conditions at infinity is investigated by the nonlinear steepest descent method of Deift and Zhou. Three asymptotic sectors in space-time plane are found: the plane wave sector I, plane wave sector II and an intermediate sector with a modulated one-phase elliptic wave. The asymptotic solutions of the three sectors are proposed by successively deforming the corresponding Riemann-Hilbert problems to solvable model problems. Moreover, a time-dependent g -function mechanism is introduced to remove the exponential growths of the jump matrices in the modulated one-phase elliptic wave sector. Finally, the modulational instability is studied to reveal the criterion for the existence of modulated elliptic waves in the central region. (C) 2018 Elsevier Inc. All rights reserved.