Solitary waves for dispersive equations with Coifman-Meyer nonlinearities (Q6592946)

This paper addresses a significant problem in the study of solitary waves within the context of dispersive equations involving nonlocal nonlinearities characterised by Coifman-Meyer Fourier operators.\N\NThe scientific problem tackled in this paper involves the existence of solitary-wave solutions for fully nonlocally nonlinear dispersive equations, a generalisation of classical unidirectional wave models such as the Korteweg-de Vries (KdV) and Boussinesq equations. The primary focus is on determining the conditions under which these solitary waves exist when the linear multiplier (M) is of a slightly higher order than the Coifman-Meyer nonlinear multiplier (N). This involves an in-depth analysis of the interaction between the orders of the linear and nonlinear terms, which ultimately governs the existence of such waves. Furthermore, the research delves into the nature of the solitary waves, analogous to those observed in KdV-type equations and capillary water waves, aiming to discover smooth solutions of various amplitudes.\N\NTo address this problem, the author employs a modified version of Weinstein's constrained minimisation argument within the calculus of variations framework. The method relies on examining the nonlinear pseudo-differential equation obtained from the evolution equation through a solitary-wave ansatz. The existence of solitary waves is then formulated as a variational problem. This involves minimising a functional representing the Hamiltonian energy under a constraint associated with the wave's amplitude, with the wave speed emerging as a Lagrange multiplier. The study also explores two specific types of Coifman-Meyer symbols, revealing that cyclical symmetry in these symbols is a necessary condition for forming a functional formulation of the problem. The concentration-compactness principle is applied to deal with the non-compactness issues arising in the nonlocal equation.\N\NThe main findings include the identification of conditions under which solitary-wave solutions exist for a given range of the parameters that define the orders of the linear and nonlinear multipliers. The research establishes that solitary waves can be found for any size of amplitude as long as the linear operator's order remains sufficiently larger than that of the nonlinear one. In particular, the study provides an existence theorem for solitary-wave solutions, demonstrating that for every positive amplitude, there is a corresponding smooth solitary-wave solution in the Sobolev space \(H^\infty(\mathbb{R})\). Furthermore, the paper gives explicit estimates for the wave speed and the amplitude, showing that the wave speed remains subcritical under the condition that the linear multiplier's order is positive and sufficiently higher than the Coifman-Meyer nonlinear multiplier's order.\N\NThe significance of this research lies in its contribution to the understanding of nonlocal nonlinear dispersive wave equations, extending the existence theory to cases with bilinear Fourier operators. By establishing analogues to classical solitary-wave theories for these more complex equations, the paper opens up new avenues for modelling various physical phenomena in dispersive media, such as water waves with surface tension. This work sets a foundation for future exploration of nonlocal nonlinearities, offering insights into the structure and behaviour of solitary waves in these advanced mathematical models.