Asymptotic analysis for rarefaction problem of the focusing nonlinear Schrödinger equation with discrete spectrum

IF 1.4 3区 物理与天体物理 Q3 PHYSICS, MATHEMATICAL
Deng-Shan Wang, Dinghao Zhu
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引用次数: 0

Abstract

The long-time asymptotic behaviors of the rarefaction problem for the focusing nonlinear Schrödinger equation with discrete spectrum are analyzed via the Riemann–Hilbert formulation. It is shown that for the rarefaction problem with pure step initial condition there are three asymptotic sectors in time–space: the plane wave sector, the 1-phase elliptic wave sector and the vacuum sector, while for the rarefaction problem with general initial data there are five asymptotic sectors in time–space: the plane wave sector, the sector of plane wave with soliton transmission, the sector of plane wave with phase shift, the sector of 1-phase elliptic wave with phase shift and the vacuum sector with phase shift. The leading-order term of each sector along with the corresponding error estimate is given by adopting the Deift–Zhou nonlinear steepest-descent method for Riemann–Hilbert problems. The asymptotic solutions match very well with the results from Whitham modulation theory and the direct numerical simulations.

Abstract Image

Abstract Image

离散谱聚焦非线性Schrödinger方程稀疏问题的渐近分析
利用黎曼-希尔伯特公式分析了具有离散谱的聚焦非线性Schrödinger方程的稀疏问题的长时间渐近行为。结果表明,对于具有纯阶跃初始条件的稀疏问题,在时空上存在三个渐近扇区:平面波扇区、一相椭圆波扇区和真空扇区,而对于具有一般初始数据的稀疏问题,在时空上存在五个渐近扇区:平面波扇区、有孤子传输的平面波扇区、有相移的平面波扇区、有相移的1相椭圆波扇区和有相移的真空扇区。采用Riemann-Hilbert问题的Deift-Zhou非线性最陡下降法,给出了各扇区的首阶项及相应的误差估计。渐近解与Whitham调制理论和直接数值模拟结果吻合较好。
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来源期刊
Letters in Mathematical Physics
Letters in Mathematical Physics 物理-物理:数学物理
CiteScore
2.40
自引率
8.30%
发文量
111
审稿时长
3 months
期刊介绍: The aim of Letters in Mathematical Physics is to attract the community''s attention on important and original developments in the area of mathematical physics and contemporary theoretical physics. The journal publishes letters and longer research articles, occasionally also articles containing topical reviews. We are committed to both fast publication and careful refereeing. In addition, the journal offers important contributions to modern mathematics in fields which have a potential physical application, and important developments in theoretical physics which have potential mathematical impact.
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