2020/09/25 by Yan Rybalko, Dmitry Shepelsky · 6 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Bar (unit) #Boundary value problem #Combinatorics #Constant (computer programming) #Function (biology) #Group (periodic table) #Initial value problem #Integrable system #Invariant (physics) #Mathematical analysis #Mathematical physics #Mathematics #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Nonlinear system #Physics #Quantum mechanics #math-ph #math.AP #math.MP
paper · pdf · doi:10.15407/mag16.04.418
published in Zurnal matematiceskoj fiziki analiza geometrii 16(4), 418-453
openalex publication_date 2020/09/25 · arxiv created 2020/11/04 · arxiv updated 2021/06/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The present paper deals with the long-time asymptotic analysis of the initial value problem for the integrable defocusing nonlocal nonlinear Schr"odinger equation iqt(x,t)+qxx(x,t)-2 q2(x,t)q(-x,t)=0 with a step-like initial data: q(x,0)→ 0 as x→ -∞ and q(x,0)→ A as x→ +∞. Since the equation is not translation invariant, the solution of this problem is sensitive to shifts of the initial data. We consider a family of problems, parametrized by R>0, with the initial data that can be viewed as perturbations of the "shifted step function" qR,A(x): qR,A(x)=0 for x<R and qR,A(x)=A for x>R, where A>0 and R>0 are arbitrary constants. We show that the asymptotics is qualitatively different in sectors of the (x,t) plane, the number of which depends on the relationship between A and R: for a fixed A, the bigger R, the larger number of sectors. Moreover, the sectors can be collected into 2 alternate groups: in the sectors of the first group, the solution decays to 0 while in the sectors of the second group, the solution approaches a constant (varying with the direction x/t=const).