Their last league outing ended in a 28-14 loss to Munster at Thomond Park and the South African has taken the decision to resign.
Despite their current standing Solomons has overseen a positive upturn in form and consecutive 1872 Scottish Cup victories over Glasgow Warriors.
"I have decided to step down. I understand Scottish Rugby have appointed Duncan Hodge as the new Acting Head Coach and I wish him well," the 66-year-old said. .
"I have greatly enjoyed working with the players and coaches and feel I have helped to put the club on a much stronger footing over the past three years and am leaving it in a better place than when I arrived."
Edinburgh legend Hodge - who played 150 times for Edinburgh over 11 seasons - rejoined Edinburgh in November 2015 after a three years coaching the national side.
Scottish Rugby Union chief executive Mark Dodson was full of praise for 26-cap Scotland international Hodge.
"Duncan brings his recent coaching experience from the Scotland national team as well as his long playing career with Edinburgh into the role and he will receive our full support to keep moving the club forward.
"I would like to take this opportunity to thank Alan for his significant contribution to the Edinburgh club and I wish him well for the future."
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