  We prove existence and conditional energetic stability of solitary-wave
solutions for the two classes of pseudodifferential equations $
u_t+\left(f(u)\right)_x-\left(L u\right)_x=0 $ and $
u_t+\left(f(u)\right)_x+\left(L u\right)_t=0, $ where $f$ is a nonlinear term,
typically of the form $c|u|^p$ or $cu|u|^{p-1}$, and $L$ is a Fourier
multiplier operator of positive order. The former class includes for instance
the Whitham equation with capillary effects and the generalized Korteweg-de
Vries equation, and the latter the Benjamin-Bona-Mahony equation. Existence and
conditional energetic stability results have earlier been established using the
method of concentration-compactness for a class of operators with symbol of
order $s\geq 1$. We extend these results to symbols of order $0<s<1$, thereby
improving upon the results for general operators with symbol of order $s\geq 1$
by enlarging both the class of linear operators and nonlinearities admitting
existence of solitary waves. Instead of using abstract operator theory, the new
results are obtained by direct calculations involving the nonlocal operator
$L$, something that gives us the bounds and estimates needed for the method of
concentration-compactness.
