The Asymptotic Behavior of the Solution of Boundary Value Problems for the sine-Gordon Equation on a Finite Interval

In this article we use the Fokas transform method to analyze boundary value problems for the sine-Gordon equation posed on a finite interval. The representation of the solution of this problem has already been derived using this transform method. We interchange the role of the independent variables to obtain an equivalent representation which can be used to study the asymptotic behavior for large times. We use this analysis to prove that the solution corresponding to constant boundary data is dominated for large times by the underlying similarity solution.