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

We study the long-time asymptotic behavior of the focusing Fokas-Lenells (FL) equation u(xt) + alpha beta(2)u - 2i alpha beta u(x) - alpha u(xx) - i alpha beta(2)vertical bar u vertical bar(2)u(x) = 0 with generic initial data in a Sobolev space which supports bright soliton solutions. The FL equation is an integrable generalization of the well-known Schrodinger equation, and also linked to the derivative Schrodinger model, but it exhibits several different characteristics from theirs. (i) The Lax pair of the FL equation involves an additional spectral singularity at k= 0. (ii) Four stationaryphase points will appear during asymptotic analysis, which require a more detailed necessary description to obtain the long-time asymptotics of the focusing FL equation. Based on the Riemann-Hilbert problem for the initial value problem of the focusing FL equation, we show that inside any fixed time-spatial cone C(x(1),x(2),nu(1),nu(2)) = {(x, t) is an element of R-2 (vertical bar)x = x(0) + nu t, x(0)is an element of[x(1), x(2)], nu is an element of [nu(1), nu(2)]}, the long-time asymptotic behavior of the solution u(x, t) for the focusing FL equation can be characterized with an N(I)-soliton on discrete spectrums and a leading order term O(t(-1/2)) on continuous spectrum up to a residual error order O(t(-3/4)). The main tool is a.-generalization of the Deift-Zhou nonlinear steepest descent method. (c) 2021 Elsevier Inc. All rights reserved.