Remarks on solitary waves in equations with nonlocal cubic terms

Abstract

In this overview paper, we show existence of smooth solitary-wave solutions to the nonlinear, dispersive evolution equations of the form ∂tu + ∂x(Λ s u + uΛ r u 2 ) = 0, where Λ s , Λ 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.