9 Result(s)
within
Panayotis G. Kevrekidis
Mathematical and Computational Engineering
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Chapter
Dispersive Shock Waves
Shock waves are nonlinear structures characterized by a discontinuous jump in the wave profile. There exist prototypical examples of partial differential equations (PDEs), including the well-known Burgers’ equ...
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Chapter
Discrete (Dark) Breathers
As indicated in the introductory chapter, discrete breathers are one of the most prototypical excitations that can arise in nonlinear dynamical lattices. They are periodic in time and strongly localized in spa...
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Chapter
Media with Onsite Forces: The Newton’s Cradle and Beyond
In this section, we finally return to the Newton’s cradle shown in Fig. 1 and Fig. 28a. When the chain of balls is struck at one end, the ball on the opposite end is ejected from the chain, providing a nice il...
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Chapter
Introduction and Motivation of Models
In the present short volume our aim will be to explore a variety of nonlinear wave structures that are found in a particular physical system, i.e., granular crystals. Many of the readers have almost certainly ...
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Chapter
Higher Dimensional Lattices
Two-dimensional granular crystals have been investigated far less than 1D configurations. In two spatial dimensions, the nodes can be arranged in a number of ways. While the role of disorder is of particular i...
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Chapter
Solitary Waves in the Nonlinear Dirac Equation
In the present work, we consider the existence, stability, and dynamics of solitary waves in the nonlinear Dirac equation. We start by introducing the Soler model of self-interacting spinors, and discuss its l...
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Chapter
Traveling Waves
Arguably, the most prototypical nonlinear wave structure that can arise in granular crystals consists of traveling waves. While many kinds of traveling waves exist (such as periodic ones [1]), in this chapter we ...
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Chapter
Heterogeneous Media
Of the three primary structures of interest in this book, solitary waves and breathers have been studied in heterogeneous granular chains, while dispersive shock waves have not. The latter represents an area o...
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Article
Dynamics of Phase Transitions in a Piecewise Linear Diatomic Chain
We consider a diatomic chain with nearest neighbors connected by phase-transforming springs. Assuming a piecewise linear interaction force, we use the Fourier transform to construct exact traveling wave soluti...
