Unifying Classical Integrable Systems via Lagrangians and Compactifications

n this research, I explore integrable systems (in the PDE sense) such as the KdV and KP systems using a geometric approach grounded in Lagrangian Multiforms. Instead of a single Lagrangian-a volume form defined over a manifold-each integrable system is associated with multiple Lagrangian volume forms (of sorts) that reflect the tower of conserved quantities inherent to these systems.

It is well known that integrable systems can be reduced dimensionally into one another. The classic example is reducing KP waves to KdV waves, essentially taking a “slice” of the KP solution. To see this, consider that while KdV solutions are roughly of the form u(x, t) = sech²(x – vt), KP solutions can be viewed as u(x, y, t) = sech²(x – vt + δy) for some δ. This idea generalizes to higher dimensions-though usually only four are needed to capture nearly all integrable systems. In the KP-to-KdV reduction, the higher-dimensional system is often identified with self-dual Yang-Mills (SDYM) theory, where introducing a translation Killing vector (symmetry) leads directly to the KP equation.

Lagrangian multiforms come into play in a similarly high-energy context. By initially considering integrable systems in the discrete setting (where they are more tractable), one considers an integrable system by taking the simplest nontrivial tower state of the system (a PDE) on a sublattice topologically equivalent to a subset of ℤⁿ. This method allows for the geometric manipulation of the submanifold, yielding reductions such as KP to KdV-in other words, reducing k-form Lagrangian multiforms to (k–1)-form ones. Thanks to the principles of multidimensional consistency and the structure of Lagrangian multiforms, one can then take this reduced single element and see it reduces the entire hierarchy consistenly. 

In this talk, I will present the technical details of this approach, including a possible look to a continuous formulation of the theory. Furthermore, I will discuss how physicists and mathematicians alike can use this formulation and more generally how lagrangian multiforms is not just a formal mathematical approach, but how it is useful in encoding both symmetries and integrability, akin to combining both the lagrangian and hamiltonian into a "single object".