Solitary waves with prescribed speed on infinite lattices
type: Item
instance of: scholarly article
title: Solitary waves with prescribed speed on infinite lattices
zbMATH Open document ID: 0889.34059
published in: Journal of Functional Analysis
publication date: 1997-12-09T00:00:00Z
review text: Using a variant of the mountain pass theorem, the authors prove the existence of solitary waves with prescribed speed on infinite lattices of particles with nearest neighbour interaction. The problem is to solve a second-order forward-backward differential-difference equation in the form \[ c^2u''= V' (u(t+1)-u(t))- V' (u(t)-u(t-1)). \tag{1} \] The following results are proven in the paper: 1. Under appropriate assumptions if \(u\) is a critical point of \[ \int_{R} {{c^2}\over 2}(u'(t))^2-V(u(t+1)-u(t)), \] then \(u\) is a solution of (1). 2. Under appropriate assumptions for every sufficiently large \(c\) the equation (1) has non-trivial non-decreasing solutions from the space \(\{u\in H^1_{loc} (R) : u' \in L^2(R),u(0) = 0\}\). 3. Under appropriate assumptions for every sufficiently large \(c\) the equation (1) has non-trivial non-increasing solutions from the space \(\{u\in H^1_{loc} (R) : u' \in L^2(R),u(0) = 0\}\).
Mathematics Subject Classification ID: 34K99
Mathematics Subject Classification ID: 70K99
zbMATH DE Number: 1078500
zbMATH Keywords: solitary waves
zbMATH Keywords: differential-difference equation
zbMATH Keywords: mountain pass theorem
author: Didier Smets
author: Michel Willem
reviewed by: Angel B. Dishliev
MaRDI profile type: Publication
full work available at URL: jfan.1996.3121
OpenAlex ID: W2073794722
cites work: Existence and numerical approximation of periodic motions of an infinite lattice of particles
cites work: Periodic motions of an infinite lattice of particles with nearest neighbor interaction
cites work: Multibump periodic motions of an infinite lattice of particles
cites work: Variational methods for indefinite superlinear homogeneous elliptic problems
cites work: Removable singularities for nonlinear elliptic equations
cites work: Existence theorem for solitary waves on lattices
cites work: On the lowest eigenvalue of the Laplacian for the intersection of two domains
cites work: On periodic motions of lattices of Toda type via critical point theory
cites work: Theory of nonlinear lattices.
cites work: Minimax theorems
DOI: 10.1006/JFAN.1996.3121
Recommended article: Existence theorem for solitary waves on lattices
Recommended article: Periodic and solitary travelling waves in infinite lattices without Ambrosetti-Rabinowitz condition
Recommended article: Approximations to solitary waves on lattices. II. quasi-continuum methods for fast and slow waves
Recommended article: Solitary waves in two-dimensional nonlinear lattices
Recommended article: Solitary wave propagation in a two-dimensional lattice
Recommended article: Solitary waves and their linear stability in nonlinear lattices
Recommended article: Q4852709
Recommended article: Solitary waves in a diatomic lattice: Analytic approximations for a wide range of speeds by quasi-continuum methods
Recommended article: Solitary waves in FPU-type lattices
Recommended article: Approximations of solitary waves on lattices using weak and variational formulations
description: scientific article; zbMATH DE number 1078500