In the present study, we consider a system of two asymmetrically coupled linear wave equations with nonhomogeneous terms. This is a part of a generalized Davey-stewartson (GDS) system which can be reduced to the Davey-Stewartson (DS) equation through a nonlinear variable transformation for some special parameters values. The GDS system has been derived to model (2+1) dimensional wave propagation in a bulk medium composed of an elastic material with coupled stresses. This system consists of one nonlinear Schrödinger (NLS) type equation for the complex amplitude of a short wave coupled with two linear wave equations for long waves propagating in the medium. The GDS system can be classified with respect to the parameter values. This classification is based on the eigenvalues of the coefficient matrix of a first order linear system with four equations equivalent to the second order linear system. The GDS system in the elliptic-hyperbolic-hyperbolic (EHH) case can be written in a dimensionless form as iμt + γ μxx + μ yy = χ ∣ μ ∣2 μ + b ( α Φ l,x + Φ2,y )μ , Φ l,xx - Φl,yy - β Φ2,xy = α ( ∣ μ ∣ 2 ) x, Φ 2,xx - ƛ ∣ μ ∣ 2 μ + b ( α Φ l,x + Φ2,y ) μ , Φ l,xx - Φ l,yy - β Φ2,xy = α ( ∣ μ ∣ 2 ) x, Φ2,xx - ƛ Φ2,yy - βΦl,xy = ( ∣ μ ∣ 2 ) ywhere x and y are spatial coordinates and t is time; u is the complex amplitude of the short transverse wave mode, and Φ1 and Φ2 are the long longitudinal and long transverse wave modes, respectively. γ, χ, b, β, α and ƛ also are constants. The purpose of the present study is to obtain integral representations of solutions to the coupled linear nonhomogeneous wave equations. Our argument is a modification of the method which is used to find an integral representation of a solution for a single wave equation. This method is that the wave equation is written in characteristic coordinates and integrations are performed along them. In order to obtain the integral representation of the solution Φ1 and Φ2 , the coupled wave system is transformed to a system of first order equations and can be solved directly. The coefficient matrix of the new system has four real distinct eigenvalues, which enables us to define the following four characteristic coordinates ξ 1 = -r1 x + y, ξ -r1 x + y η 2 = r2 x + y. Since the system does not decouple in characteristic coordinates, there is no natural pair of characteristics. On the other hand, we can either choose the characteristic pair ξ1, ξ2) , or (η1, η2) to define a coordinate transformation from the xy - pane to a plane formed by a set of the characteristic variables. In this study, we will choose the pair (ξ 1, ξ 2) and express ( η 1, η 2) in terms of (ξ 1, ξ 2).Because of the hyperbolic nature of the system, the radiation type homogeneous boundary conditions on Φ1 and Φ2 will also be assumed lim Φ1 (x,y) = lim Φ1 (x,y) = 0 and lim Φ2 (x,y) = lim Φ2 (x,y) = 0. Than we perform integrations along characteristics and make use of the radiation boundary values along the characteristics to determine the unknown functions Φ1 Φ2 . After long calculations, we obtain the representation of the solutions Φ1 Φ2 for r1 ≠ r2. We are able to see the complex interactions among the four characteristics in the limits of the integrals appearing in the representation of solutions. As a special case, when r1 = r2 , the system reduced to the decoupled system because of β = 0. Thus, solutions Φ1 Φ2 are the same as the solution for the single wave equation. In conclusion, the GDS system can be reduced to a non-local NLS equation by using the representation of solution to the coupled wave equations. This equation allows us to present some localized solutions to the GDS system for some special choices of the parameters and find some estimates of the solutions to the system. Moreover; it also helps us to investigate the global existence of the solution to the GDS system for small initial data, blow up of the solutions, asymptotic behavior of the solutions and the local existence in time of solutions without smallness condition on the data. [ABSTRACT FROM AUTHOR]