Consider the fractional KdV equation: $$ u_t + L_\alpha u_x + u u_x = 0,$$ where $L_\alpha$ denotes the Fourier multiplier operator with symbol $m(k) = |k|^\alpha$, $\alpha < -1$. I'm interest in $2\pi$-periodic traveling-wave solution $u(t,x) = \phi(x - ct)$, where $c > 0$ is the speed of the right-propagating wave. In this framework, the above equation becomes $$ -c \phi + L_\alpha \phi + \frac{1}{2} \phi^2 = B,$$ where $B \in \mathbb{R}$ is an integration constant.
Claim: Since the symbol of $L_\alpha$ is homogeneous, any bounded solution of the above equation has necessarily zero mean.
I'm not sure what this claim means. Does it mean that if $\phi$ is a $2\pi$-periodic solution, then we must have: $$\frac{1}{2\pi} \int_{-\pi}^{\pi} \phi(x) \,dx = 0 \;?$$ How can I prove it? I tried to put the equation in Fourier transform, but it did not help much.
Thank you.