Periodic finite-band solutions to the focusing nonlinear Schrödinger equation by the Fokas method: inverse and direct problems

IF 2.9 3区 综合性期刊 Q1 MULTIDISCIPLINARY SCIENCES
Dmitry Shepelsky, Iryna Karpenko, Stepan Bogdanov, Jaroslaw E. Prilepsky
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引用次数: 0

Abstract

We consider the Riemann–Hilbert (RH) approach to the construction of periodic finite-band solutions to the focusing nonlinear Schrödinger (NLS) equation. An RH problem for the solution of the finite-band problem has been recently derived via the Fokas method (Deconinck et al. 2021 Lett. Math. Phys. 111, 1–18. (doi:10.1007/s11005-021-01356-7); Fokas & Lenells. 2021 Proc. R. Soc. A 477, 20200605. (doi:10.1007/s11005-021-01356-7)) Building on this method, a finite-band solution to the NLS equation can be given in terms of the solution of an associated RH problem, the jump conditions for which are characterized by specifying the endpoints of the arcs defining the contour of the RH problem and the constants (so-called phases) involved in the jump matrices. In our work, we solve the problem of retrieving the phases given the solution of the NLS equation evaluated at a fixed time. Our findings are corroborated by numerical examples of phases computation, demonstrating the viability of the method proposed.

用福卡斯方法求聚焦非线性薛定谔方程的周期有限带解:逆问题和直接问题
我们考虑用黎曼-希尔伯特(RH)方法来构建聚焦非线性薛定谔方程(NLS)的周期性有限带解。最近通过 Fokas 方法推导出了有限带问题求解的 RH 问题(Deconinck et al.Math.111, 1-18.(doi:10.1007/s11005-021-01356-7); Fokas & Lenells.2021 Proc.R. Soc. A 477, 20200605.(doi:10.1007/s11005-021-01356-7))在此方法的基础上,NLS 方程的有限带解法可以通过相关 RH 问题的解来给出,其跃迁条件可以通过指定定义 RH 问题轮廓的弧的端点和跃迁矩阵中涉及的常数(即所谓的相位)来描述。在我们的研究中,我们解决了根据固定时间评估的 NLS 方程解找回相位的问题。我们的研究结果得到了相位计算实例的证实,证明了所提方法的可行性。
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来源期刊
CiteScore
6.40
自引率
5.70%
发文量
227
审稿时长
3.0 months
期刊介绍: Proceedings A has an illustrious history of publishing pioneering and influential research articles across the entire range of the physical and mathematical sciences. These have included Maxwell"s electromagnetic theory, the Braggs" first account of X-ray crystallography, Dirac"s relativistic theory of the electron, and Watson and Crick"s detailed description of the structure of DNA.
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