Stability of Traveling Waves to a Burgers Equation with 2nd-Order Nonlinear Diffusion

Mohammad Ghani


We are interested in the study of asymptotic stability for Burgers equation with second-order nonlinear diffusion. We first transform the original equation by the ansatz transformation to establish the existence of traveling wave. We further employ the energy estimate under small perturbation and arbitrary wave amplitude. This energy estimate is then used to establish the stability.


Stability; Arbitrary Wave Amplitude; Small Perturbation; Energy Estimate

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A. M. Il’in and O. A. Oleinik, “Asymptotic behavior of solutions of the Cauchy problem for certain quasilinear equations for large time (in Russian),” Mat. Sb., vol. 51(93), no. 2, pp. 191–216, 1960.

D. Sattinger, “On the stability of waves of nonlinear parabolic systems,” Advances in Mathematics, vol. 22, no. 3, pp. 312–355, dec 1976, doi: 0001-8708(76)90098-0.

L. Debnath, Nonlinear Partial Differential Equations for Scientists and Engineers. Boston: Birkhauser, 1997.

P. M. Jordan, “A Note on the Lambert W-function: Applications in the mathematical and physical sciences,” Contemporary Mathematics, vol. 618, pp. 247–263, 2014.

G. B. Whitham, Linear and Nonlinear Waves. New York: Wiley-Interscience, 1974.

R. E. Mickens, “Exact finite difference scheme for an advection equation having square-root dynamics,” Journal of Difference Equations and Applications, vol. 14, no. 10-11, pp. 1149–1157, oct 2008, doi:

R. Buckmire, K. McMurtry, and R. E. Mickens, “Numerical studies of a nonlinear heat equation with square root reaction term,” Numerical Methods for Partial Differential Equations, vol. 25, no. 3, pp. 598–609, may 2009, doi:

R. E. Mickens, “Wave front behavior of traveling wave solutions for a PDE having square- root dynamics,” Mathematics and Computers in Simulation, vol. 82, no. 7, pp. 1271–1277, mar 2012, doi:

R. Mickens and K. Oyedeji, “Traveling wave solutions to modified Burgers and diffusionless Fisher PDE’s,” Evolution Equations and Control Theory, vol. 8, no. 1, pp. 139–147, 2019, doi:

T. Li and Z.-A. Wang, “Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis,” Journal of Differential Equations, vol. 250, no. 3, pp. 1310–1333, feb 2011, doi:

——, “Steadily propagating waves of a chemotaxis model,” Mathematical Biosciences, vol. 240, no. 2, pp. 161–168, dec 2012, doi:

Y. Hu, “Asymptotic nonlinear stability of traveling waves to a system of coupled Burgers equations,” Journal of Mathematical Analysis and Applications, vol. 397, no. 1, pp. 322–333, jan 2013, doi:

S. Kawashima and A. Matsumura, “Stability of shock profiles in viscoelasticity with non-convex constitutive relations,” Communications on Pure and Applied Mathematics, vol. 47, no. 12, pp. 1547–1569, dec 1994, doi:

A. Matsumura and K. Nishihara, “On the stability of travelling wave solutions of a one-dimensional model system for compressible viscous gas,” Japan Journal of Applied Mathematics, vol. 2, no. 1, pp. 17–25, jun 1985, doi:

T. Nishida, “Nonlinear Hyperbolic Equations and Related Topics in Fluid Dynamics,” in Publications Math’ematiques d’Orsay 78-02. Orsay: D’epartement de Math’ematique, Universit’e de ParisSud, 1978.


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