Topological solutions of generalized Chern-Simons equations on discrete lattice graphs
Songbo Hou
TL;DR
The paper studies a generalized Chern-Simons equation on infinite lattice graphs and proves the existence of a global topological (maximal) solution that decays to zero at infinity. It combines a bounded-domain monotone iteration from $f_0=0$ with a corresponding energy functional to obtain solutions on finite graphs, and then employs an exhaustion argument to extend these solutions to the entire lattice, establishing both existence and exponential decay with rate $\alpha=\ln\left(1+\dfrac{\lambda a}{2n}\right)$. The approach provides a robust variational framework for topological vortex-type solutions on discrete graphs and yields a maximal solution on $\mathbb{Z}^n$ with explicit decay estimates. These results extend discrete Chern-Simons theory to infinite graphs and offer tools for analyzing topological structures in lattice systems.
Abstract
We study a class of generalized Chern-Simons equations on discrete lattice graphs and establish the existence of topological solutions. Using an iterative method starting from a trivial initial function and an associated energy functional, we construct a monotone decreasing sequence that converges to a solution on bounded domains. By deriving uniform estimates and passing to the limit over an increasing sequence of expanding domains, we obtain a global solution defined on the entire graph, which exhibits topological characteristics.