Kinks of the sine-gordon equation on a wormhole

Abstract

In this thesis we study the stationary Sine-Gordon equation on a wormhole (SGWH) with parameter $a$. Specifically, we establish some results for the stationary Sine-Gordon (SG) equation in flat spacetime and its kink solution $H_{SG}$. We find the $1$-kink solution $H_a(r)$ for the SGWH equation, study its asymptotic behavior as $\abs{r}\to +\infty$ and prove that it converges quadratically to $H_{SG}$ as $a\to +\infty$. In addition, the spectrum of the linearized SGWH operator is analyzed, and we show that the first eigenvalue $\lambda_a$ converges to the SG eigenvalue $\lambda_{SG}$ at a quadratic rate on $a$. Finally, we discuss the existence of $n$-kink solutions for the SGWH equation.