TL;DR
This paper studies initial-boundary value problems for sine-Gordon and complex sine-Gordon equations in a semi-strip, providing results on Weyl function evolution, solution existence, and unboundedness in the quarter-plane.
Contribution
It introduces a method to analyze the evolution of the Weyl function and establishes uniqueness and existence results for solutions in a semi-strip setting.
Findings
Weyl function evolution is characterized for complex sine-Gordon.
Solutions can be unbounded in the quarter-plane for many cases.
A procedure to recover solutions from the Weyl function is developed.
Abstract
Initial-boundary value problems for complex sine-Gordon and sine-Gordon equations in a semi--strip are treated. The evolution of the Weyl function and a uniqueness result are obtained for complex sine-Gordon equation. The evolution of the Weyl function as well as an existence result and a procedure to recover solution are given for sine-Gordon equation. It is shown that for a wide class of examples the solutions of the sine-Gordon equation are unbounded in the quarter-plane.
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