Reconstructing wake functions using Haissinski distributions from multiple bunch charges

Accurate knowledge of wake functions is crucial in accelerator physics, serving as the cornerstone for understanding intra-bunch interactions and for controlling or mitigating instabilities that limit accelerator performance. Haissinski distributions, which describe the steady-state longitudinal bunch density, are intrinsically determined by the wake function experienced by the bunch. While these distributions are typically computed from a given wake function, we investigate the inverse problem: extracting the wake function directly from measured Haissinski distributions. In this theoretical work, we introduce a novel method to reconstruct wake functions by utilizing Haissinski distributions obtained at multiple bunch charges. By combining these profiles into an overdetermined system, we address challenges posed by the inverse problem, which is sensitive to noise and discretization errors. Here, our preliminary results suggest that the use of regularization techniques may help achieve more stable reconstructions of the wake function.