Construction of solutions and asymptotics for the sine-Gordon equation in the quarter plane

Abstract

We consider the sine-Gordon equation in laboratory coordinates in the quarter plane. The first part of the paper considers the construction of solutions via Riemann-Hilbert techniques. In addition to constructing solutions starting from given initial and boundary values, we also construct solutions starting from an independent set of spectral (scattering) data. The second part of the paper establishes asymptotic formulas for the quarter-plane solution $u(x,t)$ as $(x,t) \to \infty$. Assuming that $u(x,0)$ and $u(0,t)$ approach integer multiples of $2\pi$ as $x \to \infty$ and $t \to \infty$, respectively, we show that the asymptotic behavior is described by four asymptotic sectors. In the first sector (characterized by $x/t \geq 1$), the solution approaches a multiple of $2\pi$ as $x \to \infty$. In the third sector (characterized by $0 \leq x/t \leq 1$ and $t|x-t| \to \infty$), the solution asymptotes to a train of solitons superimposed on a radiation background. The second sector (characterized by $0 \leq x/t \leq 1$ and $x/t \to 1$) is a transition region and the fourth sector (characterized by $x/t \to 0$) is a boundary region. We derive precise asymptotic formulas in all sectors. In particular, we describe the interaction between the asymptotic solitons and the radiation background, and derive a formula for the solution's topological charge.