2019/08/18 by Yan Rybalko, Dmitry Shepelsky · 2 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Algebraic structures and combinatorial models #Cauchy problem #Constant (computer programming) #Function (biology) #Heaviside step function #Initial value problem #Integrable system #Mathematical analysis #Mathematical physics #Mathematics #Method of steepest descent #Nonlinear Schrödinger equation #Nonlinear Waves and Solitons #Nonlinear system #Physics #Quantum mechanics #math.AP #msc:35Q51 #msc:37K15 #msc:37Q55 #nlin.SI
paper · pdf · doi:10.1007/s00220-021-03941-2
published in Communications in Mathematical Physics 382(1), 87-121 (Springer Science+Business Media) · 5 figures
arxiv created 2019/08/18 · openalex publication_date 2021/02/01 · arxiv updated 2021/06/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We study the Cauchy problem for the integrable nonlocal focusing nonlinear Schrödinger (NNLS) equation iqt(x,t)+qxx(x,t)+2 q2(x,t)q(-x,t)=0 with the step-like initial data close to the ``shifted step function'' χR(x)=AH(x-R), where H(x) is the Heaviside step function, and A>0 and R>0 are arbitrary constants. Our main aim is to study the large-t behavior of the solution of this problem. We show that for R∈(((2n-1)π)/(2A),((2n+1)π)/(2A)), n=1,2,…, the (x,t) plane splits into 4n+2 sectors exhibiting different asymptotic behavior. Namely, there are 2n+1 sectors where the solution decays to 0, whereas in the other 2n+1 sectors (alternating with the sectors with decay), the solution approaches (different) constants along each ray x/t=const. Our main technical tool is the representation of the solution of the Cauchy problem in terms of the solution of an associated matrix Riemann-Hilbert problem and its subsequent asymptotic analysis following the ideas of nonlinear steepest descent method.