Soliton Transmission In Space Plasma

In ideal circumstances, propagation of nonlinear Alfven waves in plasmas can be described by the derivative nonlinear Schrodinger(DNLS) equation. It can model either low-β(ratio of kinetic energy to magnetism pressure) small-amplitude Alfven waves in plasma, or high-β large-amplitude magnetohydrodynamic waves. The DNLS equation is completely integrable. Its soliton solutions both with the vanishing boundary condi-tions(VBCs) and the non-vanishing boundary conditions(NVBCs) have been found by the inverse scattering transform(IST). Here the VBC and the NVBC correspond to Alfven waves propagating exactly parallel and oblique to the magnetic field respectively. The DNLS equation supports a rich set of soliton solutions whose corresponding magnetic fields are depicted and discussed in this thesis.Completely integrable equations are highly idealized models of practical physics system. Various physical effects violating their complete integrabilities inevitably exist. When these effects are small enough, their influences on soliton transmissions can still be studied analytically by various perturbation theories for solitons.A direct perturbation theory for the DNLS equations with VBCs was recently developed. However, in general, calculations for the adiabatic evolution of the DNLS soliton are already very complicated. A symbolic computation technique is thus developed to calculate them automatically. In this thesis, we study influences of the ohmic resistance and the third-order dispersion on the DNLS soliton with VBC by using the direct perturbation theory. At first we obtain adiabatic solutions by using symbolic computation. Then we rewrite the expression for the first-order correction into a form which can be efficiently calculated by the fast Fourier transform technique and calculate them for the cases of the ohmic resistance and the third-order dispersion. Finally, we directly simulate these problems with the Split-Step-Fourier algorithm.The perturbation theory of the DNLS soliton with NVBCs has not been developed. It is well known that completely integrable equations possess infinite number of conservation laws. In the presence of perturbations, conserved quantities also evolve with perturbations. If evolution equations for soliton parameters obtained from evolution equations are consistent, we can find adiabatic evolution equations for soliton parameters. Therefore, we find the infinite number of conservation laws for the DNLS equation with NVBC and their evolution equations under perturbations. However , we find the evolution equations for soliton parameters are inconsistent, declaring the failure of this approach for perturbed DNLS solitons with NVBC and summoning a systematic perturbation theory.