Perturbation theory for kinks. (English) Zbl 0756.35084

Summary: We prove the validity of formal asymptotic results on perturbation theory for kink solutions of the sine-Gordon equation, originally obtained by D. W. McLaughlin and A. C.Scott [Phys. Rev. A18, No. 4, 1652- 1678 (1978)]. We prove that for appropriate perturbations, of size \(\varepsilon\) in an appropriate norm, slowly varying in time in the rest frame of the kink, the shape of the kink is unaltered in the \(L^ \infty\) norm to \(O(\varepsilon)\) for a time of \(O(1/\varepsilon)\). The kink parameters, which represent its velocity and centre, evolve slowly in time in the way predicted by the asymptotics. The method of proof uses an orthogonal decomposition of the solution into an oscillatory part and a one-dimensional “zero-mode” term. The slow evolution of the kink parameters is chosen so as to suppress secular evolution of the zero- mode.


35Q53 KdV equations (Korteweg-de Vries equations)
35B20 Perturbations in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
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