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F. Achleitner, C. Cuesta, S. Hittmeir:
"Travelling waves for a non-local Korteweg-de Vries-Burgers equation";
Journal of Differential Equations, 257 (2014), 3; S. 720 - 758.

We study travelling wave solutions of a Korteweg-de Vries-Burgers equation with a non-local diffusion term. This model equation arises in the analysis of a shallow water flow by performing formal asymptotic expansions associated to the triple-deck regularisation (which is an extension of classical boundary layer theory). The resulting non-local operator is a fractional derivative of order between 1 and 2. Travelling wave solutions are typically analysed in relation to shock formation in the full shallow water problem. We show rigorously the existence of these waves. In absence of the dispersive term, the existence of travelling waves and their monotonicity was established previously by two of the authors. In contrast, travelling waves of the non-local KdV-Burgers equation are not in general monotone, as is the case for the corresponding classical KdV-Burgers equation. This requires a more complicated existence proof compared to the previous work. Moreover, the travelling wave problem for the classical KdV-Burgers equation is usually analysed via a phase-plane analysis, which is not applicable here due to the presence of the non-local diffusion operator. Instead, we apply fractional calculus results available in the literature and a Lyapunov functional. In addition we discuss the monotonicity of the waves in terms of a control parameter and prove their dynamic stability in case they are monotone.

Non-local evolution equation; Fractional derivative; Travelling waves

http://dx.doi.org/10.1016/j.jde.2014.04.015