Classical Yang-Baxter equation, Lagrangian multiforms and ultralocal integrable hierarchies

We cast the classical Yang-Baxter equation (CYBE) in a variational context for the first time, by relating it to the theory of Lagrangian multiforms, a framework designed to capture integrability in a variational fashion. This provides a significant connection between Lagrangian multiforms and the CYBE, one of the most fundamental concepts of integrable systems. This is achieved by introducing a generating Lagrangian multiform which depends on a skew-symmetric classical $r$-matrix with spectral parameters. The multiform Euler-Lagrange equations produce a generating Lax equation which yields a generating zero curvature equation. The CYBE plays a role at three levels: 1) It ensures the commutativity of the flows of the generating Lax equation; 2) It ensures that the generating zero curvature equation holds; 3) It implies the closure relation for the generating Lagrangian multiform. The specification of an integrable hierarchy is achieved by fixing certain data: a finite set $S\in CP^1$, a Lie algebra $\mathfrak{g}$, a $\mathfrak{g}$-valued rational function with poles in $S$ and an $r$-matrix. We show how our framework is able to generate a large class of ultralocal integrable hierarchies by providing several known and new examples pertaining to the rational or trigonometric class. These include the Ablowitz-Kaup-Newell-Segur hierarchy, the sine-Gordon (sG) hierarchy and various hierachies related to Zakharov-Mikhailov type models which contain the Faddeev-Reshetikhin (FR) model and recently introduced deformed sigma/Gross-Neveu models as particular cases. The versatility of our method is illustrated by showing how to couple integrable hierarchies together to create new examples of integrable field theories and their hierarchies. We provide two examples: the coupling of the nonlinear Schr\"odinger system to the FR model and the coupling of sG with the anisotropic FR model.