Stability analysis of multiple solutions of three wave interaction with group velocity dispersion and wave number mismatch

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.