A Finite - Gap Background in Bullough - Dodd - Jiber - Shabat Model

makes a good alternative to the well-known Sine-Gordon equation as one more model of the two-dimensional integrable relativistic field theory with the self-action. In [3] the authors had suggested the construction of the finite-gap solutions for this equation. These are periodic and/or almost periodic function in x, t associated with special classes of double sheet ramified coverings of Riemann surfaces and represented by explicit formulae in terms of Prym theta-functions of such coverings. In this paper we construct Hirota type (see [4]) determinant formula for N -soliton solutions of (1.1) and use it for the investigation of the space-time asymptotics of them in space-time variables where (1.1) has a form