The Crab pulsar and its pulsar-wind nebula in the optical and infrared

Aims. We investigate the emission mechanism and evolution of pulsars that are associated with supernova remnants. Methods. We used imaging techniques in both the optical and near infrared, using images with very good seeing (<= 0 ''.6) to study the immediate surroundings of the Crab pulsar. In the case of the infrared, we took two data sets with a time window of 75 days to check for variability in the inner part of the Crab nebula. We also measure the spectral indices of all these wisps, the nearby knot, and the interwisp medium, using our optical and infrared data. We then compared the observational results with the existing theoretical models. Results. We report variability in the three nearby wisps located to the northwest of the pulsar and also in a nearby anvil wisp in terms of their structure, position, and emissivity within the time window of 75 days. All the wisps display red spectra with similar spectral indices (alpha(nu) = -0.58 +/- 0.08, alpha(nu) = -0.63 +/- 0.07, alpha(nu) = -0.53 +/- 0.08) for the northwest triplet. The anvil wisp (anvil wisp 1) has a spectral index of alpha(nu) = -0.62 +/- 0.10. Similarly, the interwisp medium regions also show red spectra similar to those of the wisps, with the spectral index being alpha(nu) = -0.61 +/- 0.08, alpha(nu) = -0.50 +/- 0.10, while the third interwisp region has a flatter spectrum with spectral alpha(nu) = -0.49 +/- 0.10. The inner knot has a spectral index of alpha(nu) = -0.63 +/- 0.02. Also, based on archival HST data and our IR data, we find that the inner knot remains stationary for a time period of 13.5 years. The projected average velocity relative to the pulsar for this period is less than or similar to 8 km s(-1). Conclusions. By comparing the spectral indices of the structures in the inner Crab with the current theoretical models, we find that the Del Zanna et al. model for the synchrotron emission fits our observations, although the spectral index is at the flatter end of their modelled spectra.