Long-time asymptotics for the focusing Fokas-Lenells equation in the solitonic region of space-time

We study the long-time asymptotic behavior of focusing Fokas-Lenells (FL) equation u x t + α β 2 − i | = 0 with generic initial data in a Sobolev space which supports bright soliton solutions. The FL is an integrable generalization well-known Schrodinger equation, and also linked to derivative model, but it exhibits several different characteristics from theirs. (i) Lax pair involves additional spectral singularity at k . (ii) Four stationary phase points will appear during analysis, require more detailed necessary description obtain asymptotics equation. Based on Riemann-Hilbert problem for value we show that inside any fixed time-spatial cone C ( 1 , v ) { ∈ R [ ] } solution can be characterized N I -soliton discrete spectrums leading order term O / continuous spectrum up residual error 3 4 main tool ∂ ‾ -generalization Deift-Zhou nonlinear steepest descent method.