Two-dimensional simulations, both Vlasov and particle-in-cell, are presented that show the evolution of the field and electron distribution of finite-width, nonlinear electron plasma waves. The intrinsically intertwined effects of self-focusing and dissipation of field energy caused by electron trapping are studied in simulated systems that are hundreds of wavelengths long in the transverse direction but only one wavelength long and periodic in the propagation direction. From various initial wave states, both the width at focus ${\Delta }_m$ relative to the initial width ${\Delta }_0$ and the maximum field amplitude at focus are shown to be a function of the growth rate of the transverse modulational instability ${\gamma }_{\mathrm{TPMI}}$ divided by the loss rate of field energy ${\nu }_E$ to electrons escaping the trapping region. With dissipation included, an amplitude threshold for self-focusing ${\gamma }_{\mathrm{TPMI}}/{\nu }_E\sim 1$ is found that supports the analysis of Rose [Phys. Plasmas 12, 012318 (2005)].

• Received 9 June 2013

DOI:https://doi.org/10.1103/PhysRevLett.111.105002