Nonlinear diffusive shock acceleration with upstream escape reproduces DAMPE observations

We develop a self-consistent nonlinear extension of diffusive shock acceleration that incorporates cosmic ray (CR) backreaction on the shock precursor together with a physically motivated upstream-escape mechanism that produces an exponential high energy cutoff. The CR pressure gradient decelerates the upstream flow facing the shock wave, generating an extended precursor in which higher rigidity particles sample a larger cumulative velocity gradient and thereby acquire a progressively harder spectrum. Finite-size/escape effects are modeled by a momentum-dependent loss term, which naturally terminates acceleration and steepens the spectrum near the cutoff. The precursor compression ratio is not imposed as a closure condition but is determined dynamically by requiring consistency between the injection rate inferred from thermal leakage at the subshock and the injection strength demanded by the nonlinear shock modification, with CR-driven wave heating providing stabilizing negative feedback. Applying the model to young supernova-remnant-like parameters and standard one-zone Galactic diffusion, we reproduce the main features of the latest DAMPE proton spectrum: gradual hardening from hundreds of GeV to multi-TeV energies and a subsequent exponential cutoff at tens of TeV. The resulting spectral evolution follows directly from the competition between precursor-mediated nonlinear feedback and upstream escape.