"Nonlinear Waves in Disordered Diatomic Granular Chains"

Every scientist has written papers that have interesting stories. Like everybody else, I have several of these, and my paper that was published in final form today is one such paper. You can see the title of this blog entry, and I will give the abstract and some other comments below, but let me first tell you a story.

I have been working on nonlinear waves in granular crystals since November 2006, and this has become the main component of my nonlinear waves research. It's been a fun and challenging adventure and I am looking forward to continuing in it. The idea for this paper arose from three different occurrences: a question from an audience member in the seminar that I gave to OCIAM (the research group I'm in) when I first joined the faculty in Fall 2007, a seminar I gave the same term at University of Cambridge, and a similar idea by e-mail when I was showing a collaborator (from a different set of projects) a draft of a much earlier paper of mine on granular crystals. In each of these cases, I was asked about the possibility of Anderson localization in disordered granular crystals.

I hadn't thought about it before nor did I know too much about Anderson localization (though I had heard about it), but it sounded interesting, and eventually my collaborators and I designed an undergraduate student project to study disordered granular crystals. The student project started in summer 2008 and eventually became this paper, which was first submitted to a journal in April 2009 and had a bit of a rough path. The first referees asked us to remove language related to Anderson localization, and the fifth and final referee (five---count 'em---five referees, though I do actually have one paper that needed 6 referees) asked us to put that stuff back in. [In fact, that referee all but called one of the previous ones an idiot.] It got a bit frustrating at times, but the published version of the paper is so much better than the original version, so in many senses the pain was worth it. And now the paper is finally out!

OK, so what did we find? Well, I was asked several times about the possibility of Anderson localization in granular crystals. What actually occurs is a different and seemingly novel form of localization, and in my view our paper opens up the problem of what exactly this phenomenon is. I have no idea what kind of impact this paper will ultimately have and I have several papers in more prestigious journals, but I do feel like my collaborators and I have opened up a pretty damned interesting problem with this paper. Alex, I'll take "nonlinear localization" for the win!

Oh, and here is the formal paper information:

Title: Nonlinear Waves in Disordered Diatomic Granular Chains

Authors: Laurent Ponson, Nicholas Boechler, Yi Ming Lai, Mason A. Porter, P. G. Kevrekidis, and Chiara Daraio

Abstract: We investigate the propagation and scattering of highly nonlinear waves in disordered granular chains composed of diatomic (two-mass) units of spheres that interact via Hertzian contact. Using ideas from statistical mechanics, we consider each diatomic unit to be a "spin," so that a granular chain can be viewed as a spin chain composed of units that are each oriented in one of two possible ways. Experiments and numerical simulations both reveal the existence of two different mechanisms of wave propagation: in low-disorder chains, we observe the propagation of a solitary pulse with exponentially decaying amplitude. Beyond a critical level of disorder, the wave amplitude instead decays as a power law, and the wave transmission becomes insensitive to the level of disorder. We characterize the spatiotemporal structure of the wave in both propagation regimes and propose a simple theoretical interpretation for a transition between the two regimes. Our investigation suggests that an elastic spin chain can be used as a model system to investigate the role of heterogeneities in the propagation of highly nonlinear waves.