Sorry to be pedantic but the term 'instrumental song' is an oxymoron. Songs have words. The term you want is 'violin pieces'.
My favourites are the Brahms Sonatas for violin and piano.
Mm - thumbs down - how mature.
There is no such thing as an "instrumental song" - by definition if it is instrumental it is not a song. If you are looking for melancholy music the suggestion of Schubert is a good one. He wrote both instrumental and vocal music. Start by listening to his piano music - the impromptus and the late piano sonatas.
I love all the Vivaldi concerto, I have never heard one yet that I didn't like, my favorite is the "The Four Seasons". I also love all the transcribed music of Bach's for violin !
"Meditation" is a good one...
Pieces from Vivaldi will be great too...
He composed many pieces for violin solos and duets...
If you have a string quartet, I'll recomend "Assez Vif" by Maurice Ravel...