Initial Conditions and Korteweg-de Vries Solitons

by Patrick D. Weidman, Assoc. Prof. of Mechanical Engrg.; Univ. of Colorado, Boulder, Colo.,
Larry G. Redekopp, Prof. of Aerospace Engrg.; Univ. of S. Calif., Los Angeles, Calif.,

Serial Information: Journal of the Engineering Mechanics Division, 1982, Vol. 108, Issue 2, Pg. 277-289

Document Type: Journal Paper


A study is made of the asymptotic waveforms described by the Korteweg-de Vries equation (ut + 6uux = uxxx = 0) evolving from initial data composed of discrete disturbances of elevation and depression. The analytical approach is based on the inverse scattering transform which relates the initial condition u(x,0) to Schrodinger's equation. Solution of direct scattering problem for an initial disturbance composed of three contiguous rectangles is obtained. The poles of the reflection coefficient determine the number of solitons in the asymptotic state, and specific results are obtained for the following types of initial data: two contiguous rectangles; three contiguous rectangles which are symmetric in x; and two distinct separated rectangles. The competing effect of disturbances of elevation (which give rise to solitons) and depression (which give rise to dispersive waves) is studied in detail. The results can be summarized by stating that the number of evolved solitons depends on the strength of each rectangular disturbance, the relative amplitudes of the rectangular disturbances, and the relative proximity of the disturbances.

Subject Headings: Symmetry

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