2012/09/01 by E. Ben-Naim, E Ben-Naim, P L Krapivsky +1 · 1 citation
Mathematics · Physics and Astronomy · #Burgers' equation #Exponential function #Focus (optics) #Fractional Differential Equations Solutions #Heuristic #Logarithm #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Ordinary differential equation #Set (abstract data type) #Traveling wave #cond-mat.stat-mech #nlin.PS
paper · pdf · doi:10.1088/1751-8113/45/45/455003
published as J. Phys. A 45, 455003 (2012) · 6 pages, 5 figures
arxiv created 2012/09/01 · openalex publication_date 2012/10/29 · arxiv updated 2014/12/25 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We propose the set of coupled ordinary differential equations dnj/dt=(nj-1)2-(nj)2 as a discrete analog of the classic Burgers equation. We focus on traveling waves and triangular waves, and find that these special solutions of the discrete system capture major features of their continuous counterpart. In particular, the propagation velocity of a traveling wave and the shape of a triangular wave match the continuous behavior. However, there are some subtle differences. For traveling waves, the propagating front can be extremely sharp as it exhibits double exponential decay. For triangular waves, there is an unexpected logarithmic shift in the location of the front. We establish these results using asymptotic analysis, heuristic arguments, and direct numerical integration.