We propose a novel closed-form time domain (TD) Volterra series nonlinear equalizer (VSNE) for the mitigation of Kerr-related distortions in polarization-multiplexed (PM) coherent optical transmission systems. The proposed TD-VSNE is obtained from inverse Fourier analysis of a frequency domain (FD) VSNE based on a frequency-flat approximation. Employing novel TD approximations, we demonstrate the equivalency between the VSNE algorithms formulated in time and frequency domains. In order to enhance the computational efficiency, we insert a power weighting time window in the TD-VSNE, yielding the weighted VSNE (W-VSNE) algorithm. We demonstrate that the convergence of the W-VSNE to its maximum performance is much faster than that of the TD-VSNE, thus requiring fewer parallel filters. Through numerical simulation of a 224 Gb/s PM- 16QAM optical channel, we compare the performance/complexity tradeoff of the W-VSNE with the well known split-step Fourier method (SSFM) and with the computationally optimized weighted SSFM (W-SSFM). Enabled by the use of fewer iterations and only 2 parallel W-VSNE filters, we demonstrate a reduction of up to 45% on computational effort and 70% on latency, in comparison with the W-SSFM.