Exact 1D Nonlinear Solutions for Proton-Driven Plasma Wakefields

Abstract

We extend the classical 1D nonlinear relativistic plasma wakefield paradigm to evaluate the pristine fluid profiles driven by intense relativistic proton trains under the comoving coordinate approximation. Using an intense two-bunch scheme, our adaptive integration model is validated down to machine precision against exact closed-form cold-fluid invariants, confirming that interface boundary corners represent physical step discontinuities in the second potential derivative. Scaling to an $N=100$ micro-bunch train, we resolve macroscopic phase-accumulation and subsequent nonlinear detuning by deriving a closed-form, parameter-free piecewise analytical envelope expression. This solution successfully maps the initial unattenuated quadratic coherent growth alongside the flat asymptotic horizontal saturation horizon governed by nonlinear energy conservation. Furthermore, the framework analytically evaluates the transient envelope overshoot, tracking its decay onto the saturation boundary via an exact phase-slip integration length scale. Providing immediate, ready macro-envelope bounds, this computationally inexpensive tool eliminates empirical adjustments and yields a robust mathematical baseline designed to guide and accelerate complex, multi-dimensional particle-in-cell (PIC) simulations through rapid parameter space exploration before deploying heavy computational assets. It establishes an efficient foundation to optimize asymmetric, custom-shaped driver profiles required to break the fundamental symmetric transformer ratio limit of 2 to make current particle accelerators more efficient.

arXiv commentSubmitted for publication, revised version, Python script used to generate figure 1 is openly available in Zenodo at https://doi.org/10.5281/zenodo.23492771