论文标题
孤子与气体：动力学方程背后的弗雷霍尔姆决定因素，分析和快速振荡
Soliton versus the gas: Fredholm determinants, analysis, and the rapid oscillations behind the kinetic equation
论文作者
论文摘要
我们分析了一个密集的MKDV孤子气体及其在单个试验孤子存在下的较大时间行为。我们表明，该解决方案可以用弗雷德尔姆的决定因素以及Riemann-Hilbert问题来表达。然后，我们表明该解决方案可以分解为背景气体溶液的总和（调制椭圆波），以及孤子解决方案：但是，由于相互作用动力学，单个表达式非常复杂。此外，我们能够在孤子通过后得出气体的局部相移，并且随着动力学的发展，我们可以追踪孤子峰的位置。最后，我们表明，在与孤子气体相互作用的同时，孤子峰具有振荡速度，其领先顺序平均值满足类似于V. Zakharov和G. El所提出的动力学速度方程。
We analyze the case of a dense mKdV soliton gas and its large time behaviour in the presence of a single trial soliton. We show that the solution can be expressed in terms of Fredholm determinants as well as in terms of a Riemann-Hilbert problem. We then show that the solution can be decomposed as the sum of the background gas solution (a modulated elliptic wave), plus a soliton solution: the individual expressions are however quite convoluted due to the interaction dynamics. Additionally, we are able to derive the local phase shift of the gas after the passage of the soliton, and we can trace the location of the soliton peak as the dynamics evolves. Finally we show that the soliton peak, while interacting with the soliton gas, has an oscillatory velocity whose leading order average value satisfies the kinetic velocity equation analogous to the one posited by V. Zakharov and G. El.
