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Travelling-wave solutions to the Korteweg-de Vries-Burgers equation. (English) Zbl 0594.76015

R. S. Johnson [J. Fluid Mech. 42, 49-60 (1970; Zbl 0213.549)] suggested that the equation (1)\(u_ t+uu_ x+\delta u_{xxx}- \epsilon u_{xx}=0\) may serve as an approximate model for waves in physical systems in which the weak effects of nonlinearity, dissipation, and dispersion are present. In this paper, consideration was given to the travelling-wave solutions (2)u(x,t)\(=S(x-ct\); \(\epsilon\),\(\delta)\), where c is a fixed positive constant. This paper has been organized as follows. Section 2 is devoted to establishing the existence and uniqueness of the solutions (2), corresponding to given positive values of c, \(\epsilon\) and \(\delta\). A more detailed description of the travelling-wave solution is given in section 3. Certain qualitative aspects of these solutions are proved. Sections 4 and 5 delve into the limiting behavior of S(\(\cdot;\epsilon,\delta)\) as \(\epsilon\) \(\to 0\) and as \(\delta\) \(\to 0\), respectively. In section 6, the limiting form of S(\(\cdot;\epsilon,\delta)\) as \(\epsilon\) and \(\delta\) simultaneously tend to zero is examined. In the case where the ratio \(\delta /\epsilon^ 2\) remains bounded it transpires that S converges to a shock-wave profile.
Reviewer: Lung Y.Shih

MSC:

76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35Q99 Partial differential equations of mathematical physics and other areas of application

Citations:

Zbl 0213.549
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References:

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