Traveling waves in Fermi-Pasta-Ulam chains with nonlocal interaction

Alexander Pankov

doi:10.3934/dcdss.2019135

Article Contents

2019, Volume 12, Issue 7: 2097-2113. Doi: 10.3934/dcdss.2019135

This issue Previous Article Periodic and subharmonic solutions for a 2$n$th-order $\phi_c$-Laplacian difference equation containing both advances and retardations Next Article Ground states of nonlinear Schrödinger equations with fractional Laplacians

Traveling waves in Fermi-Pasta-Ulam chains with nonlocal interaction

Alexander Pankov

Mathematics Department, Morgan State University, Baltimore, MD 21251, USA

Dedicated to Professor Norman Dancer on the occasion of his 70th birthday

Received: May 2018

Revised: June 2018

Published: June 2019

The author is supported by Simons Foundation, award 410289.

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Abstract

The paper is devoted to traveling waves in FPU type particle chains assuming that each particle interacts with several neighbors on both sides. Making use of variational techniques, we prove that under natural assumptions there exist monotone traveling waves with periodic velocity profile (periodic waves) as well as waves with localized velocity profile (solitary waves). In fact, we obtain periodic waves by means of a suitable version of the Mountain Pass Theorem. Then we get solitary waves in the long wave length limit.

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Mathematics Subject Classification: Primary: 37K60, 37K40; Secondary: 34A33, 34C25, 74J35, 58E05.

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