| Popis: |
In the first part of this thesis algebro-geometric solutions for the sine-Gordon and KdV equations in the presence of a type I integrable defect are found, generalising the previously known soliton solutions. Elliptic (genus one) solutions where the defect induces only a phase shift are obtained via ansätze for the fields on each side of the defect. Algebro-geometric solutions for arbitrary genus and involving soliton emission by the defect are constructed using a Darboux transformation, exploiting the fact that the defect equations have the form of a Bäcklund transformation at a point. All the soliton and phase-shifted elliptic solutions to the defect equations are recovered as limits of the algebro-geometric solutions constructed in this way. Certain energy and momentum conserving defects for the Kadomtsev-Petviashvili equation are then presented as a first step towards the construction of integrable defects in higher dimensions. Algebro-geometric solutions to the sine-Gordon equation on the half-line with an integrable two parameter boundary condition are obtained by imposing a corresponding restriction on the Lax pair eigenfunction or, alternatively, as a Darboux transformation of the known algebro-geometric solution for the Dirichlet boundary. Finally, the collision of sine-Gordon solitons with a Robin type boundary is examined. This boundary is typically non-integrable but becomes an integrable Neumann or Dirichlet boundary for certain values of a boundary parameter. Depending on the boundary parameter and initial velocity an antikink may be reflected into various combinations of kinks, antikinks and breathers. The soliton content of the field after the collision is numerically determined by computing the discrete scattering data associated with the inverse scattering method. A highlight of this investigation is the discovery of an intricate structure of resonance windows caused by the production of a breather which can collide multiple times with the boundary before escaping as a lighter breather or antikink. |
