I have the following dataframe (sim_2005):
Date ELEM1 ELEM2 ... ELEM1133 2005-01-01 0.021 2.455 ... 345.2 2005-01-02 0.321 2.331 ... 355.1 ...
Assuming Date is the index to your DataFrame, you can get a date column in your melted DataFrame as follows:
Date
sim_2005_melted['Date'] = pd.concat([sim_2005.reset_index().Date for _ in range(sim_2005.shape[1])], ignore_index=True).values