Wobbling kinks in φ4 and sine-Gordon theory

Abstract

When the φ⁴ model admits a kink solution, it also admits a wobbling kink, which satisfies the boundary conditions of a kink, but possesses an internal degree of freedom. In this paper we develop a formal perturbation series for the wobbling kink in φ⁴ theory, and give the first two terms in the series explicitly. Then we prove that the formal series actually is asymptotic for a rather long time [O(K ln(1/ε)) for a certain K]. Finally, we construct an exact 3‐soliton solution of the sine‐Gordon equation that also has the properties of a wobbling kink. For the sine‐Gordon equation, the wobbling kink seems to be mildly unstable.