Asymptotic stability for monotone traveling kinks of the dissipative Boussinesq problem

We study the dissipative Boussinesq problem, which extends the “good” Boussinesq equation by incorporating viscosity effects. It is well-known that this model supports monotone decreasing traveling kink solutions. We show that these kinks are orbitally stable in $H^1(\text{rone})$. Moreover, they are asymptotically stable in ${\dot{H}}^1(\text{rone})$ and $L^p(\text{rone}),2<p\le \mathrm{\infty }$.