用Deift-Zhou方法求无限远处具有非零边界条件的聚焦Hirota方程的长时间渐近性

IF 0.9 3区数学 Q3 MATHEMATICS, APPLIED

Mathematical Physics, Analysis and Geometry Pub Date : 2021-05-08 DOI:10.1007/s11040-021-09388-0

Shuyan Chen, Zhenya Yan, Boling Guo

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引用次数: 8

摘要

我们关注聚焦Hirota方程(也称为三阶非线性Schr?在无穷远处具有对称的非零边界条件(nzbc)的dinger方程。首先，基于具有nzbc的Lax对，利用正散射和逆散射问题建立了具有不同跳跃曲线的振荡Riemann-Hilbert (RH)问题。其次，采用Deift-Zhou非线性最陡下降法分析了振动性RH问题，并分别在两个不同的时空平面域(平面波域和调制椭圆波域)上给出了长时间渐近解。最后，还研究了所考虑的Hirota方程的调制不稳定性。

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Long-Time Asymptotics for the Focusing Hirota Equation with Non-Zero Boundary Conditions at Infinity Via the Deift-Zhou Approach

We are concerned with the long-time asymptotic behavior of the solution for the focusing Hirota equation (also called third-order nonlinear Schr?dinger equation) with symmetric, non-zero boundary conditions (NZBCs) at infinity. Firstly, based on the Lax pair with NZBCs, the direct and inverse scattering problems are used to establish the oscillatory Riemann-Hilbert (RH) problem with distinct jump curves. Secondly, the Deift-Zhou nonlinear steepest-descent method is employed to analyze the oscillatory RH problem such that the long-time asymptotic solutions are proposed in two distinct domains of space-time plane (i.e., the plane-wave and modulated elliptic-wave domains), respectively. Finally, the modulation instability of the considered Hirota equation is also investigated.

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来源期刊

Mathematical Physics, Analysis and Geometry 数学-物理：数学物理

CiteScore

2.10

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： MPAG is a peer-reviewed journal organized in sections. Each section is editorially independent and provides a high forum for research articles in the respective areas. The entire editorial board commits itself to combine the requirements of an accurate and fast refereeing process. The section on Probability and Statistical Physics focuses on probabilistic models and spatial stochastic processes arising in statistical physics. Examples include: interacting particle systems, non-equilibrium statistical mechanics, integrable probability, random graphs and percolation, critical phenomena and conformal theories. Applications of probability theory and statistical physics to other areas of mathematics, such as analysis (stochastic pde''s), random geometry, combinatorial aspects are also addressed. The section on Quantum Theory publishes research papers on developments in geometry, probability and analysis that are relevant to quantum theory. Topics that are covered in this section include: classical and algebraic quantum field theories, deformation and geometric quantisation, index theory, Lie algebras and Hopf algebras, non-commutative geometry, spectral theory for quantum systems, disordered quantum systems (Anderson localization, quantum diffusion), many-body quantum physics with applications to condensed matter theory, partial differential equations emerging from quantum theory, quantum lattice systems, topological phases of matter, equilibrium and non-equilibrium quantum statistical mechanics, multiscale analysis, rigorous renormalisation group.