Orbital stability of solitary waves of the Schrödinger-Boussinesq equation

This paper concerns the orbital stability of solitary waves of the Schrödinger-Boussinesq equation $ i\partial_t u+\partial_x^2 u+uv =0\qquad\qquad\qquad\qquad\qquad (0.1)$ $ \partial_t^2 v-\partial_x^2 v+\partial_x^4 v+\partial_x^2 (3v^2+|u|^2)=0.$ By applying the abstract results of Grillakis, Shatah and Strauss [11, 12] and detailed spectral analysis developed by Lopes in [17], we obtain the stability of solitary waves.