  We present the quantum phase transition in two capacitively coupled arrays of
superconducting quantum dots (SQD). We consider the presence of gate voltage in
each superconducting island. We show explicitly that the co-tunneling process
involves with two coupled SQD arrays, near the maximum charge frustration line
is not sufficient to explain the correct quantum phases with physically
consistent phase boundaries. We consider another extra co-tunneling process
along each chain to explain the correct quantum phases with physically
consistent phase boundaries. There is no evidence of supersolid phase in our
study. We use Bethe-ansatz and Abelian bosonization method to solve the problem
