TWO-PHASE ASYMPTOTICS IN THE FORM OF AIRY FUNCTIONS IN THE WAVE BREAKING PROBLEM FOR THE I-BURGERS EQUATION
Abstract
We consider wave breaking problems for the Burgers equation with a small “imaginary viscosity,” which in fact plays the role of small dispersion. Although this equation has no apparent physical meaning, the problem in question is an interesting analog of the famous Gurevich-Pitaevsky problem on the onset of an oscillation zone as the breaking of a simple wave occurs for the Korteweg-de Vries equation. In contrast to the latter, the solution of the i-Burgers equation in the oscillation zone can be described explicitly and has a two-phase structure. This was indicated more than 25 years ago in (Dobrokhotov et al., 1992), where the solution was constructed in the form of a function of Maslov’s canonical operator. Now we use the recent results in (Dobrokhotov, Nazaikinskii, 2018) to present the solutions in the more efficient form of uniform asymptotics represented as the logarithmic derivative of the Airy function of a composite argument.
The research was supported by the Ministry of Science and Higher Education of the Russian Federation (project no. AAAA-A17-117021310377-1).
References
- Dobrokhotov S.Yu., Maslov V.P. and Tsvetkov V.B. Zadacha ob oprokidyvanii volny dlya model’nogo uravneniya vt + vvx - ih2vxx= 0 (Problem of the reversal of a wave for the model equation vt + vvx - ih2vxx= 0 ). Math. Notes, 1992, Vol. 51, No. 6, pp. 624–627.
- Dobrokhotov S.Yu. and Nazaikinskii V.E. Efficient formulas for the canonical operator near a simple caustic. Russ. J. Math. Physics, 2018, Vol. 25, No. 3, pp. 545–553.
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