Remarks on solitary waves in equations with nonlocal cubic terms

In this overview paper, we show existence of smooth solitary-wave solutions to the nonlinear, dispersive evolution equations of the form \begin{equation*} \partial_t u + \partial_x(\Lambda^s u + u\Lambda^r u^2) = 0, \end{equation*} where $\Lambda^s, \Lambda^r$ are Bessel-type Fourier multipliers. The linear operator may be of low fractional order, $s>0$, while the operator on the nonlinear part is assumed to act slightly smoother, $r<s-1$. The problem is related to the mathematical theory of water waves; we build upon previous works on similar equations, extending them to allow for a nonlocal nonlinearity. Mathematical tools include constrained minimization, Lion's concentration-compactness principle, spectral estimates, and product estimates in fractional Sobolev spaces.