Publication 25-CNA-010

Traveling Wave Profiles for a Semi-discrete Burgers Equation

Uditnarayan Kouskiya
Department of Civil and Environmental Engineering
Vanderbilt University
Nashville, 37212, TN, USA
uditnarayan.kouskiya@vanderbilt.edu

Robert L. Pego
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
rpego@cmu.edu

Amit Acharya
Dept. of Civil & Environmental Engineering
Center for Nonlinear Analysis
Carnegie Mellon University
Pittsburgh, PA 15213
acharyaamit@cmu.edu

Abstract: We look for traveling waves of the semi-discrete conservation law $4 \dot {u}_j + u_{j+1}^2 - u_{j-1}^2 = 0$, using variational principles related to concepts of "hidden convexity" appearing in recent studies of various PDE (partial differential equations). We analyze and numerically compute with two variational formulations related to dual convex optimization problems constrained by either the differential-difference equation (DDE) or nonlinear integral equation (NIE) that wave profiles should satisfy. We prove existence theorems conditional on the existence of extrema that satisfy a strict convexity criterion, and numerically exhibit a variety of localized, periodic and nonperiodic wave phenomena.

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