Classical Integrable Field Theories with Defects and near-Integrable Boundaries
Mar 26, 2018158 pages
Supervisor:
Thesis: PhD - York U., England
- Published: Mar 26, 2018
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Abstract: (submitter)
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. Algebrogeometric
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.- field theory: integrability
- defect: integrability
- sine-Gordon equation: solution
- Korteweg-de Vries equation: solution
- soliton
- Darboux transformation
- Kadomtsev-Petviashvili equation
- boundary condition
- kink
- inverse scattering method
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