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Traveling wave profiles for a semi-discrete Burgers equation

Uditnarayan Kouskiya, Robert L. Pego, Amit Acharya

TL;DR

This work addresses traveling wave profiles in the nonintegrable, semi-discrete Burgers equation $4\dot u_j + u_{j+1}^2 - u_{j-1}^2 = 0$ by developing two dual variational formulations—one from the differential-difference equation and another from the associated nonlinear integral equation. Central to the approach is a dual-to-primal mapping and the notion of hidden convexity, enabling constrained concave maximization on adaptively chosen bases to compute wave profiles, including localized solitary waves, periodic waves, and dispersive envelopes. The authors prove conditional existence results tied to convexity constraints and provide extensive numerical evidence of diverse wave phenomena, including solitary, dispersive, and oscillatory-decay profiles, as well as instances where Petviashvili iterations fail to converge. The study highlights regime-dependent behaviors controlled by phase-speed non-matching conditions and offers a flexible computational framework that can adapt bases to navigate nonlocal constraints, with implications for nonintegrable discretizations of conservation laws. Overall, the work contributes a robust variational methodology for nonlocal traveling waves in semi-discrete systems and reveals rich wave dynamics beyond integrable analogues, suggesting directions for rigorous existence proofs and broader applications.

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 non-periodic wave phenomena.

Traveling wave profiles for a semi-discrete Burgers equation

TL;DR

This work addresses traveling wave profiles in the nonintegrable, semi-discrete Burgers equation by developing two dual variational formulations—one from the differential-difference equation and another from the associated nonlinear integral equation. Central to the approach is a dual-to-primal mapping and the notion of hidden convexity, enabling constrained concave maximization on adaptively chosen bases to compute wave profiles, including localized solitary waves, periodic waves, and dispersive envelopes. The authors prove conditional existence results tied to convexity constraints and provide extensive numerical evidence of diverse wave phenomena, including solitary, dispersive, and oscillatory-decay profiles, as well as instances where Petviashvili iterations fail to converge. The study highlights regime-dependent behaviors controlled by phase-speed non-matching conditions and offers a flexible computational framework that can adapt bases to navigate nonlocal constraints, with implications for nonintegrable discretizations of conservation laws. Overall, the work contributes a robust variational methodology for nonlocal traveling waves in semi-discrete systems and reveals rich wave dynamics beyond integrable analogues, suggesting directions for rigorous existence proofs and broader applications.

Abstract

We look for traveling waves of the semi-discrete conservation law , 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 non-periodic wave phenomena.

Paper Structure

This paper contains 24 sections, 5 theorems, 142 equations, 41 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Variational formulations
  3. Dual DDE formulation
  4. Dual NIE Formulation
  5. Analysis of concave maximization problems
  6. Analysis for the dual DDE formulation
  7. Analysis for the dual NIE formulation
  8. Approximation and numerical examples for the DDE formulation
  9. Approximation for the DDE formulation
  10. A modified Newton-Raphson scheme with step-size control
  11. Numerical examples for the DDE formulation
  12. Approximating solutions with prior knowledge
  13. Approximating solutions without any prior knowledge
  14. Dispersive solitons and their disintegrations
  15. Approximation and numerical examples for the NIE formulation
  16. ...and 9 more sections

Key Result

Proposition 3.1

Let $0<L<\infty$ and let $\bar{f}\in L^2(\Omega_1)$. The functional $\tilde{S}$ defined by eq:sup_inf is given by eq:tilde_S and maps $H^1_0(\Omega)$ into $[-\infty,\frac{1}{2}\mathsf{a}\|\bar{f}\|^2]$. Moreover, $\tilde{S}$ is concave and upper semicontinuous. The interior of its domain $\mathop{\m Furthermore, $-\tilde{S}$ is coercive, and $\tilde{S}$ achieves a maximum. If some maximizer $\lamb

Figures (41)

  • Figure 1: Base state: $\mathcal{PV}$ solution
  • Figure 2: Base state: $2\,\mathcal{PV}$ solution
  • Figure 4: Single hump primal profiles
  • Figure 5: Double hump profile
  • Figure 7: $\mathsf{a} = 1$
  • ...and 36 more figures

Theorems & Definitions (9)

  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • Proposition 3.5
  • proof