Context :

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 ( \leq 0 . ^ { \prime \prime } 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 \pm 0.08 , \alpha _ { \nu } = -0.63 \pm 0.07 , \alpha _ { \nu } = -0.53 \pm 0.08 ) for the northwest triplet .
The anvil wisp ( anvil wisp 1 ) has a spectral index of \alpha _ { \nu } = -0.62 \pm 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 \pm 0.08 , \alpha _ { \nu } = -0.50 \pm 0.10 , while the third interwisp region has a flatter spectrum with spectral \alpha _ { \nu } = -0.49 \pm 0.10 .
The inner knot has a spectral index of \alpha _ { \nu } = -0.63 \pm 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 \phantom { \leq } \smash { \buildrel \over { \lower 2.675022 pt \hbox { $ \buildrel% \lower 2.140017 pt \hbox { $ \displaystyle < $ } \over { \sim } $ } } } 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 .
( 2006 ) model for the synchrotron emission fits our observations , although the spectral index is at the flatter end of their modelled spectra .