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The process of photon-photon scattering in vacuum is investigatedanalytically in the long-wavelength limit within the framework of theEuler-Heisenberg Lagrangian. In order to solve the nonlinear partialdifferential equations (PDEs) obtained from this Lagrangian use is made of thehodograph transformation. This transformation makes it possible to turn asystem of quasilinear PDEs into a system of linear PDEs. Exact solutions of theequations describing the nonlinear interaction of electromagnetic waves invacuum in a one-dimensional configuration are obtained and analyzed.

Hodograph solutions of the wave equation of nonlinear electrodynamics in the quantum vacuum