Solitons Induced by Boundary Conditions.
Although soliton phenomena have attracted wide attention since 1965, there are still not enough efforts paid to mixed boundary-initial value problems which are important in real physical cases. The main purpose of this thesis is to study carefully the various boundary induced solitons under different initial conditions. We start with three sets of nonlinear equations: KdV equation and Boussinesq equation (for water); two fluid equations for the cold ion plasma. We have been interested in four types of problems involved with water solitons: excitation by different time -dependent boundary conditions under different initial conditions; head-on and over-taking collisions; reflection at a wall and the excitation by pure initial conditions. For KdV equation, only cases one and four are conducted. The results from two fully non-linear KdV and Boussinesq equations are compared, and agree extremely well. In both sets of equations, for the constant boundary but finite (nonzero) unperturbed water level, the ratio between the disturbed amplitude and the unperturbed water level plays the key role in determining the soliton maturing time. When this ratio is small, new solitons take a long time to mature; results of 23 are held at the opposite extremes. Dealing with various time-dependent boundary conditions, the ratio between the soliton maturing period and the time-scale of boundary controls the envelope of solitons induced. A random boundary for KdV equation (unperturbed water level approaches zero) can also induce a series of solitons as a constant boundary does. This is a very striking result. The Boussinesq equation permits solition head -on collisions and reflections, studied the first time. The results from take-over collision agree with KdV results. The numerical schemes used are up to fourth order accurate in x and second order accurate in t. For the KdV equation, the schemes from 23 are adopted. For the ion acoustic plasma, we have derived a set of Boussinesq-type equations from the standard two fluid equations for the ion acoustic plasma. It theoretically proves the essential nature of the solitary wave solutions of the cold-ion plasma. The ion acoustic solitons are also obtained by prescribing a potential (phi)(,0) at one grid point. A part of these results had been presented in the 1984 APS Plasma Physics Division Meeting and the IMACS 1985 World Congress.
- Pub Date:
- Physics: Fluid and Plasma