Stability of travelling waves to Benjamin-Ono type equations with double power nonlinearities

We study the stability of travelling wave solutions to dispersion generalized Benjamin--Ono type equations with the double power nonlinearities |u|ᵖ⁻¹u + |u|q-1u \ (1 < p < q < ∞). We prove that travelling wave solutions constructed by ground states are stable if p is L²-subcritical and speeds of waves are sufficiently small by showing the coercivity of the linearized operator around ground states, which is introduced by Grillakis-Shatah-Strauss (1987).