Stability and instability of solitary waves in fractional generalized KdV equation in all dimensions

Abstract

We study stability of solitary wave solutions for the fractional generalized Korteweg-de Vries equation $$ \partial_t u- \partial_{x_1} D^{\alpha}u+ \tfrac{1}{m}\partial_{x_1}(u^m)=0, ~ (x_1,\dots,x_d)\in \mathbb{R}^d, \, \, t\in \mathbb{R}, \, \, 0<\alpha <2, $$ in any spatial dimension $d\geq 1$ and nonlinearity $m>1$. The arguments developed here are independent of the spatial dimension and rely on the new estimates for spatial decay of ground states and their regularity. In the $L^2$-subcritical case, we prove the orbital stability of solitary waves using the concentration-compactness argument, the commutator estimates and expansions of nonlocal operator $D^\alpha$ in several variables. In the $L^2$-supercritical case, we show that solitary waves are unstable. More precisely, the instability is obtained by constructing an explicit sequence of initial conditions that move away from a soliton orbit in finite time, this is shown in conjunction with the modulation and truncation arguments, and incorporating the decay and regularity of the ground states. As a consequence, in 1D we show the instability of solitary waves of the supercritical generalized Benjamin-Ono equation ($\alpha=1$) and the dispersion-generalized Benjamin-Ono equation ($1<\alpha<2$); furthermore, new results on the instability are obtained in the weaker dispersion regime when $\frac{1}{2}<\alpha<1$. This work should be of interest in studying stability of solitary waves and other coherent structures in a variety of dispersive equations that involve nonlocal operators.