Is Android has specific file transfer protocol via Wifi Direct?
WiFi direct is a communication protocol just like Bluetooth. What differentiates between the two bluetooth has a transfer protocol; Object Exchange (OBEX) that facilitates the exchange of binary objects between devices) while the former doesn't.
AFAIK there is no standard protocol for file sharing even in Android environment.
WiFi direct is just one layer in a whole stack of different communication technologies and protocols required to get file sharing over network connections to function.
Basically you need to layer transport and session protocols layers on top (like TCP/IP), additionally you need to layer application protocols on top of TCP/IP, (like FTP, NFS, CIFS)
what is the best way for file transfer between Linux and Android systems?
Well it depends, perhaps using USB connection? you're guaranteed realibity and speed. Sometimes we don't need to complicate things.
Wireless methods? (i.e bluetooth, wi-fi Wi-Fi direct etc) but again other factors come into play: compatibility, session protocols (FTP, CIFS etc) some protocols are associated with high speeds and reliability e.g cifs> ftp etc.
See one example of such scenario using wi-fi: File transfer with linux pc through mobile hotspot
WIFI Direct is now enabled in wp8.1-- devs can we expect file sharing apps using wifi direct APIs???