Wobbling kinks in ϕ4 and sine-Gordon theory
When the ϕ4 model admits a kink-solution, it also admits a wobbling king, 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 rahter 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.
KeywordsFormal Series Secular Term Discrete Eigenvalue Kink Solution Inverse Scattering Transform
Unable to display preview. Download preview PDF.
- M. J. Rice, Phys. Lett. 71A, 152, (1979).Google Scholar
- D. K. Campbell and A. R. Bishop, Nuclear Physics B 200, 297, (1982).Google Scholar
- D. K. Campbell, J. F. Schonfeld, and C. A. Wingate, Resonance structure in kin-antikink interaction in ϕ 4 theory. (Preprint.)Google Scholar
- M. J. Ablowitz, and H. Segur, Solitons and the Inverse Scattering Transform. SIAM, Philadelphia, 1981.Google Scholar
- G. L. Lamb, Elements of Soliton Theory. Wiley-Interscience, New York, 1980.Google Scholar
- I. M. Gel'fand and B. M. Levitan, American Mathematical Society Translation, Series 2, 1, 259, (1955).Google Scholar
- G. G. Stokes, Trans. Cambridge Phil. Soc. 8, 81 (1847); (Papers 1, 197).Google Scholar