Classical and quantum dynamics of a charged scalar particle in a background of two counterpropagating plane waves

We consider a scalar particle in a background formed by two counterpropagating plane waves. Two cases are studied: (i) dynamics at a magnetic node and (ii) zero initial transverse canonical momentum. The Lorentz and Klein-Gordon equations are solved for these cases and approximations analyzed. For the magnetic-node solution (homogeneous, time-dependent electric field), the modified Volkov wave function which arises from a high-energy approximation is found to be inaccurate for all energies and the solution itself unstable when the photon emission (nonlinear Compton scattering) is included. For the zero initial transverse canonical momentum case, in both quantum and classical cases, forbidden parameter regimes, absent in the plane-wave model, are identified.