Reply to Comment on "Existence of Internal Modes of sine-Gordon Kinks"

TL;DR

This paper clarifies misconceptions about the existence of internal modes in sine-Gordon kinks, reaffirming previous results that quasimodes do not exist, by addressing and correcting earlier claims and assumptions.

Contribution

The authors refute claims that internal quasimodes exist in sine-Gordon kinks, reaffirming their previous findings through clarification of simulation conditions and critique of prior proofs.

Findings

Simulations were conducted with kinks having nonzero initial velocity.

Previous proof assuming static kinks was shown to be an approximation, not rigorous.

The existence of internal quasimodes in the sine-Gordon equation remains unsupported.

Abstract

In this reply to the comment by C. R. Willis, we show, by quoting his own statements, that the simulations reported in his original work with Boesch [Phys. Rev. B 42, 2290 (1990)] were done for kinks with nonzero initial velocity, in contrast to what Willis claims in his comment. We further show that his alleged proof, which assumes among other approximations that kinks are initially at rest, is not rigorous but an approximation. Moreover, there are other serious misconceptions which we discuss in our reply. As a consequence, our result that quasimodes do not exist in the sG equation [Phys. Rev. E 62, R60 (2000)] remains true.