Remarks on solitary waves in equations with nonlocal cubic terms

TL;DR

This paper demonstrates the existence of smooth solitary-wave solutions for a class of nonlinear dispersive equations with nonlocal cubic terms, extending previous water wave models to include nonlocal nonlinearities.

Contribution

It introduces new existence results for solitary waves in equations with nonlocal cubic nonlinearities, utilizing advanced mathematical tools like concentration-compactness and spectral estimates.

Findings

Existence of smooth solitary-wave solutions proven.

Extension of water wave theory to nonlocal nonlinearities.

Application of fractional Sobolev space techniques.

Abstract

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.