(Received January 21, 1966)
In this paper we consider a model of the dispersive non-linear equation which is derived from the Schrödinger equation by means of a non-linear transformation. The derivation is closely analogous to deriving the Burgers model from the diffusion equation. Then we obtain a solution resulting from an initial discontinuity, which corresponds to that of Riemann's problem in the Burgers model. In the solution so obtained, the limit that the Planck constant vanishes is discussed in comparison with the dissipative limit in the Burgers model which is realized by letting the dissipation constant tend to zero. It is explicitly shown that although in these limits the equations in the two models are the same, nevertheless their solutions are entirely different; for example, as is well known, in the Burgers model one discontinuity develops from the initial discontinuity while in our model two discontinuities develop, across each of which the so-called generalized Rankine-Hougoniot relations are not valid.
URL : http://ptp.ipap.jp/link?PTP/35/1142/
DOI : 10.1143/PTP.35.1142