Zero-dispersion limit to the Korteweg-de Vries equation: a dressing chain approach
- 33 Downloads
An asymptotic solution of the KdV equation with small dispersion is studied for the case of smooth hump-like initial condition with monotonically decreasing slopes. Despite the well-known approaches by Lax-Levermore and Gurevich-Pitaevskii, a new way of constructing the asymptotics is proposed using the inverse scattering transform together with the dressing chain technique developed by A. Shabat . It provides the Whitham-type approximaton of the leading term by solving the dressing chain through a finite-gap asymptotic ansatz. This yields the Whitham equations on the Riemann invariants together with hodograph transform which solves these equations explicitly. Thus we reproduce an uniform in x asymptotics consisting of smooth solution of the Hopf equation outside the oscillating domain and a slowly modulated cnoidal wave within the domain. Finally, the dressing chain technique provides the proof of an asymptotic estimate for the leading term.
Key wordsKdV small dispersion limit wave collapse dressing chain
MSC2000 numbers34E15 35Q53 35Q51 37K15
Unable to display preview. Download preview PDF.
- 3.Gurevich, A.V., Krylov, A.L., and Mazur, N.G., Quasi-simple Waves in Korteweg-de Vries Hydrodynamics, Zh. Eksp. Teor. Fiz., 1989, vol. 95, p. 1674–1689.Google Scholar
- 6.Gurevich, A.V. and Pitaevskii, L.P., The Non-stationary Structure of a Collisionless Shock Wave, Zh. Eksp. Teor. Fiz., 1973, vol. 65, pp. 590–604. [Soviet Physics JETP, 1973, vol. 38, p. 291].Google Scholar
- 9.Zaharov, V.E., Manakov, S.V., Novikov, S.P., and Pitaevskii, L.P., Teoriya solitonov (Theory of Solitons), Moscow: Nauka, 1980.Google Scholar
- 13.Gurevich, A.V., Krylov, A.L., and El’, Riemann Wave Breaking in Dispersive Hydrodynamics, Zh. Eksp. Teor. Fiz., 1991, vol. 54, p. 102–107.Google Scholar