A MODEL OF DISPERSIVE NON-LINEAR EQUATION.
INSTITUTE OF PLASMA PHYSICS NAGOYA UNIV (JAPAN)
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In this paper we consider a model of dispersive non -linear equation, which is derived from the Schrodinger 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 the Riemanns 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 a dissipation constant tend to zero. It is shown explicitly that in these limits equations in the two models are same nevertheless their solutions are entirely different for example, as is well known in the Burgers model one discontinuity develops from the initial discontinuity whilst in our model two discontinuities develop, across each of which the so-called generalized Rankine-Hugoniot relations are not valid.
- Numerical Mathematics
- Plasma Physics and Magnetohydrodynamics