Solving nonlinear wave problems with spectral boundary value techniques.

Abstract

Solitary waves are important in modeling geophysical flows. They have been the basis for successful models of coherent structures, such as the persistent high pressure systems in the atmosphere, Gulf Stream rings, and the Great Red Spot of Jupiter. Most previous numerical studies have solved the nonlinear evolution equations as initial value problems. Here, they are approached as spectral boundary value problems. One technique used is to analytically expand the equation in powers of an amplitude parameter via the Stokes' perturbation expansion, which is accurate for small amplitude. The solution may then be extended to larger amplitude using a Newton-Kantorovich iteration. We have applied these techniques to several problems. The first problem addressed is wave resonance; the phase speeds of two wave components are equal and the traditional Stokes' expansion fails. The situation is rectified by adding the resonant wave into the expansion at the appropriate lower order. The coefficient of the resonant wave is determined at higher order in the expansion, resulting in a modified expansion which agrees well with the numerical solution. The case of near resonance is treated by expanding the resonance parameter in terms of the amplitude as well. A second problem is computing double cnoidal waves (two independent waves on each period) of the integrable Korteweg de Vries equation and the nonintegrable Regularized Long Wave (RLW) equation. The nonlinear evolution equation is written in terms of two phase dimensions and the techniques described above are applied. The KdV solution compares well to the exact solution in the soliton limit. Although we cannot prove the double cnoidal waves of the RLW exist, the results agree well with initial value solutions for moderate amplitudes.Ph.D.Physics, Atmospheric SciencePure SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/128156/2/8812905.pd