Collective coordinate description of kink-antikink interaction

Takyi, Ishmael (2015-12)

Thesis (MSc)--Stellenbosch University, 2015.

Thesis

ENGLISH ABSTRACT: In this thesis we explore resonance solutions in one space and one time dimensions in the nonlinear non-integrable ø⁴ and ø⁶ models. These resonances occur when the relative velocity between the kink and antikink falls below a specific critical velocity. It has been claimed that the presence of an internal vibrational mode is a necessary condition for the existence of these solutions. However, this mode is part of the solutions in the small amplitude formulation only in the ø⁴ model. Based on this, we investigate whether the availability of such a mode in the (reduced) collective coordinate ansätze will explain the resonance solutions regardless of whether or not it is a solution within the small amplitude approximation. We extend previous collective coordinate ansätze to three independent functions. Except for special scenarios this circumvents the problem seen in earlier studies that the coordinates are ill-declined when the kink-antikink separation tends to zero. We quantitatively compare the numerical results of the ordinary differential equations obtained from these ansätze to that of the partial differential equations in both the ø⁴ and ø⁶ models.

AFRIKAANSE OPSOMMING: In hierdie tesis ondersoek ons die resonansie oplossings in een ruimte en een tyd dimensie van die nie-lineere nie-integreebare ø⁴ en ø⁶ modelle; wat voorkom wanneer die relativistiese snelheid tussen die kink en antikink val onder 'n sekere kritiese waarde. Dit is 'n standaard veronderstelling dat die teenwoordigheid van 'n interne vibrasie mode 'n noodsaaklike kondisie vir die bestaan van hierdie vibrasie oplossings is nietemin is die oplossings ook deel van die lae amplitude formulering van die ø⁴ model. Gebaseer op hierdie idiee ondersoek ons of die beskikbaarheid van hierdie mode in die (gereduseerde) kollektiewe koordinaat ansätze die resonansie sal verduidelik ongeag of dit 'n oplossing in die klein amplitude benadering is. Ons oorweeg ook 'n kollektiewe koordinaat ansätze met drie vryheids grade in so 'n manier dat die evolusie van die resonansie veranderlike wat onstaan goed gedefinieer is. Ons vergelyk die numeriese resultate van die gewone differensiaal vergelykings wat volg met die volledige stel gedeeltelike differensiaal vergelykings beide nie ø⁴ en ø⁶ modelle.

Please refer to this item in SUNScholar by using the following persistent URL: http://hdl.handle.net/10019.1/97877
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