This paper concerns the orbital stability for exact solitary waves of the Generalized Klein-Gordon-Schrod-inger equations. Since the abstract results of Grillakis et al[1-2] can not be applied directly, we can extend the abstract stability theory and use the detailed spectral analysis to obtain the stability of the solitary waves.
M .Grillakis, J. Shatah and W. Strauss, “Stability Theory of Solitary Waves in the Presence of Symmetry, Ⅰ,” Journal of Functional Analysis, Vol. 74, No. 1, 1987, pp. 160-197. doi:10.1016/0022-1236(87)90044-9
J. P. Albert and J. L. Bona, “Total Positivity and the Stability of Internal Waves in Stratified Fluids of Finite Depth,” IMA Journal of Applied Mathematics, Vol. 46, No. 1-2, 1991, pp. 1-19. doi:10.1093/imamat/46.1-2.1