Nonlinear integral equations for finite volume excited-state energies of the O(3) and O(4) nonlinear σ-models

We propose nonlinear integral equations for the finite volume one-particle energies in the O(3) and O(4) nonlinear σ-models. The equations are written in terms of a finite number of components and are therefore easier to solve numerically than the infinite component excited-state TBA equations proposed earlier. Results of numerical calculations based on the nonlinear integral equations and the excited-state TBA equations agree within numerical precision.