近日，德国汉诺威大学的Oliver Melchert与Ayhan Demircan合作并取得一项新进展。经过不懈努力，他们对三阶和负四阶色散扰动下非线性薛定谔方程孤波解进行数值研究。相关研究成果已于2024年10月22日在国际知名学术期刊《物理评论A》上发表。

本文从数值研究了受三阶色散和负四阶色散效应扰动的非线性薛定谔方程的孤波解。已知在单一波数下，存在一个具有非零速度的局域化解的解析表达式，这里称之为克鲁格洛夫-哈维孤波解。为了获得一般波数和速度下的孤波，研究人员采用一种定制的谱重正化方法。

对于选定的一组系统参数和一系列波数，研究人员通过拟合模型对所得脉冲进行表征，从而能够推导出脉冲参数之间的经验关系。通过碰撞可以获得对这些孤波相互作用动力学的更深入理解。这些碰撞通常是非弹性的，并允许形成具有非常特殊动力学的短寿期双脉冲束缚态。

最后，研究人员详细阐述了克鲁格洛夫-哈维孤子解在弱损耗下的特性。对于短传播距离，这一数值结果验证了早期扰动理论的预测，并表明脉冲形状在传播过程中会发生变化。对于长距离，研究人员观察到向线性脉冲展宽的过渡。对于较短的传播距离，数值结果证实了微扰理论的早期预测，并表明脉冲的形状在传播过程中发生了改变。对于长距离，该课题组观察到线性脉冲展宽的交叉。

附：英文原文

Title: Numerical investigation of solitary-wave solutions for the nonlinear Schrodinger equation perturbed by third-order and negative fourth-order dispersion

Author: Oliver Melchert, Ayhan Demircan

Issue&Volume: 2024/10/22

Abstract: We numerically study solitary-wave solutions for the nonlinear Schrdinger equation perturbed by the effects of third-order and negative fourth-order dispersion. At a single wave number, an analytical expression for a localized solution with nonzero velocity, here referred to as Kruglov and Harvey's solitary-wave solution, is known to exist. To obtain solitary waves for general wave numbers and velocities, we employ a custom spectral renormalization method. For a selected set of system parameters and a range of wave numbers, we characterize the resulting pulses via a fit model, allowing us to formulate empirical relations between the pulse parameters. Deeper insight into the interaction dynamics of these solitary waves can be obtained through collisions. These collisions are typically inelastic and allow for the formation of short-lived two-pulse bound states with very particular dynamics. Finally, we detail the properties of Kruglov and Harvey's soliton solution under weak loss. For short propagation distances our numerical results verify earlier predictions of perturbation theory and show that the pulse shape is altered upon propagation. For long distances we observe a crossover to linear pulse broadening.

DOI: 10.1103/PhysRevA.110.043518

Source: https://journals.aps.org/pra/abstract/10.1103/PhysRevA.110.043518