  This paper explores the analytical approach for obtaining the multiple
solutions of three-wave interacting system in (1+1) dimensions. We present a
novel approach by expressing the wave solutions in terms of Jacobi elliptic
functions and delve into specific cases involving hyperbolic functions.
Additionally, the paper focuses on analysing the linear stability of two kinds
of solutions: (a) periodic and (b) one or two-hump bright solitons due to group
velocity and group velocity dispersion. For linear stability, we solve the
eigenvalue problem by Fourier collocation method, where Fourier coefficients
are defined analytically and compared numerically. On the other hand, we check
the linear stability by direct numerical simulations with Pseudospectral method
along special derivatives $(t)$ and 4th order Runge-Kutta method in temporal
direction $(z)$. Then it is confirmed by Crank-Nicholson finite difference
method. Furthermore, we introduce a special case known as constant magnitude
wave solution and examine its modulational instability in presence of group
velocity dispersion. In addition, the influence of group velocities and wave
vector mismatch are investigated.
