Solitary waves for dispersive equations with Coifman-Meyer nonlinearities

TL;DR

This paper proves the existence of solitary waves in certain nonlocal nonlinear dispersive equations using a modified Weinstein's approach, highlighting the importance of the linear multiplier's order relative to the nonlinear one.

Contribution

It introduces a new method for establishing solitary wave existence in fully nonlocal equations with Coifman-Meyer nonlinearities, emphasizing the role of the multiplier's order.

Findings

Existence of solitary waves for equations with positive, higher-order linear multipliers.

Necessary cyclical symmetry in Coifman-Meyer symbols for a variational formulation.

Wave speed and solution estimates near bifurcation points.

Abstract

Using a modified version of Weinstein's argument for constrained minimization in nonlinear dispersive equations, we prove existence of solitary waves in fully nonlocally nonlinear equations, as long as the linear multiplier is of positive and slightly higher order than the Coifman-Meyer nonlinear multiplier. It is therefore the relative order of the linear term over the nonlinear one that determines the method and existence for these types of equations. In analogy to Korteweg-De Vries-type equations and water waves in the capillary regime, smooth solutions of all amplitudes can be found. We consider two structural types of symmetric Coifman-Meyer symbols $n(\xi-\eta,\eta)$, and show that cyclical symmetry is necessary for the existence of a functional formulation. Estimates for the solution and wave speed are given as the solutions tend to the bifurcation point of solitary waves.