The Calogero model bears, in the continuum limit, collective excitations in the form of density waves and solitary modulations of the density of particles. This sector of the spectrum of the model was investigated, mostly within the framework of collective field theory, by several authors, over the past fifteen years or so. In this work we shall concentrate on periodic solutions of the collective BPS-equation (also known as "finite amplitude density waves"), as well as on periodic solutions of the full static variational equations which vanish periodically (also known as "large amplitude density waves"). While these solutions are not new, we feel that our analysis and presentation add to the existing literature, as we explain in the text. In addition, we show that these solutions also occur in a certain two-family generalization of the Calogero model, at special points in parameter space. A compendium of useful identities associated with Hilbert transforms, including our own proofs of these identities, appears in Appendix A. In Appendix B we also elucidate in the present paper some fine points having to do with manipulating Hilbert transforms, which appear ubiquitously in the collective field formalism. Finally, in order to make this paper self-contained, we briefly summarize in Appendix C basic facts about the collective field formulation of the Calogero model.
Solitons in the Calogero model for distinguishable particles
我们考虑一个大的$- N, $两族Calogero模型的哈密顿，集体场方法。利用静态孤子组态给出了Bogomol'nyi极限的出现和相应的解。来自不同族的孤子被局域在同一位置。它们的行为类似于均匀真空凝聚体顶部的成对孔和块，取决于耦合强度的值。当第一族中的粒子数远大于第二族时，空穴解会转到单族Calogero模型中已经发现的涡廓线。
Solitons on Noncommutative Torus as Elliptic Calogero Gaudin Models, Branes and Laughlin Wave Functions
对于非对易环面${\cal T}$，在N.C.参数$θ= \frac{Z}{n}$，我们构造了Hilbert空间的基${\ca$ H}_n $ in terms of $ θ $ functions of the positions $ z_i $ of $ n $ solitons. The wrapping around the torus generates the algebra $ { \cal A}_n $, which is the $ Z_n \times Z_n $ Heisenberg group on $ θ $ functions. We find the generators $ g $ of an local elliptic $ su（n）$, w$变换协变地由${\cal A}$的整体规范变换通过作用于${\cal H}_n$我们建立了${\cal A}_n$ $g$的同构.我们把这个$g$嵌入到椭圆Gaudin的$L$ -矩阵中，$models to give the dynamics. The moment map of this twisted cotangent $ su_n（{ \cal T}）$ bundle is matched to the $ D $-equation with Fayet-Illiopoulos source te…
