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 counter-propagating 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 analysed. For the magnetic node solution (homogeneous, time-dependent electric field), the modified Volkov wavefunction which arises from a high-energy approximation is found to be inaccurate for all energies and the solution itself unstable when 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.