可积Boussinesq方程的孤子气体及其广义流体力学
Soliton gas of the integrable Boussinesq equation and its generalised hydrodynamics
AI总结：
本文研究了Boussinesq方程的孤子气体，通过受Thermodynamic Bethe Ansatz启发的启发式方法构建其广义流体力学，为理解(2+1)维KP孤子气体及各向异性双向孤子气体提供了新视角。
摘要：
广义流体力学（GHD）是研究非平衡多体可积系统（量子或经典）的新近且强大的框架。它已被应用于多个模型，从delta Bose gas到XXZ spin chain、KdV孤子气体等。然而，它仅被应用于(1+1)维系统，向更高空间维度的推广并非易事。我们研究Boussinesq方程，尽管它通常被认为物理相关性不如KdV方程，但它作为（boosted）Kadomtsev-Petviashvili（KP）方程的平稳约化而颇具研究价值，而KP方程是(2+1)维非线性可积PDE的典型且普适的范例。我们采用受Thermodynamic Bethe Ansatz启发的启发式方法来构建Boussinesq孤子气体的GHD。该方法允许对Boussinesq孤子气体进行统计力学解释，这自然伴随GHD图像而来。这应被视为构建KP孤子气体的第一步，为某些类别的解提供了洞察，使我们能够建立直觉以设计更普遍的理论。这也为Congy等人（2021）先前通过唯象引入的各向异性双向孤子气体的构建提供了另一视角。
英文摘要：
Generalised hydrodynamics (GHD) is a recent and powerful framework to study many-body integrable systems, quantum or classical, out of equilibrium. It has been applied to several models, from the delta Bose gas to the XXZ spin chain, the KdV soliton gas and many more. Yet it has only been applied to (1+1)-dimensional systems and generalisation to higher dimensions of space is non-trivial. We study the Boussinesq equation which, while generally considered to be less physically relevant than the KdV equation, is interesting as a stationary reduction of the (boosted) Kadomtsev-Petviashvili (KP) equation, a prototypical and universal example of a nonlinear integrable PDE in (2+1) dimensions. We follow a heuristic approach inspired by the Thermodynamic Bethe Ansatz in order to construct the GHD of the Boussinesq soliton gas. Such approach allows for a statistical mechanics interpretation of the Boussinesq soliton gas that comes naturally with the GHD picture. This is to be seen as a first step in the construction of the KP soliton gas, yielding insight on some classes of solutions from which we may be able to build an intuition on how to devise a more general theory. This also offers another perspective on the construction of anisotropic bidirectional soliton gases previously introduced phenomenologically by Congy et al (2021).
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