Sine-Gordon kink (${\varphi }_K=4\arctan e^{x-a}$)

The static kink ${\varphi }_K(x)=4\arctan e^{x-a}$ joins adjacent scalar-field vacua $0$ and $2\pi$. It obeys ${\varphi }_K^{\prime }=2\mathrm{sech}(x-a)$ and has topological charge $Q=[\varphi (+\infty )-\varphi (-\infty )]/(2\pi )=1$. In the Sine-Gordon theory normalization with physical mass scale $m$ and coupling $\beta$, its classical mass is $8m/{\beta }^2$. A Lorentz boost gives ${\varphi }_K=4\arctan \exp [(x-vt-a)/\sqrt{1-v^2}]$. Spatial reflection gives an antikink.

This localized two-soliton field has zero net winding and describes an elastic kink-antikink collision for $0<v<1$. Two reciprocal positive Bäcklund parameters produce it from the vacuum. Analytic continuation of $v$ to an imaginary value gives a real Sine-Gordon breather. The overall field sign and spacetime translations change conventions, not the field equation.

For $0<v<1$ and $\Gamma =(1-v^2{)}^{-1/2}$, this real two-soliton solution has angular winding $4\pi$. Its separated kink velocities are $\pm v$. At large times the centers satisfy $|m\Gamma x|=|m\Gamma vt|-\log v+o(1)$, giving a right-moving shift $-2\log v/(m\Gamma )$. It follows from Bianchi permutability for sine-Gordon Bäcklund transformations with oppositely signed seed parameters.

The second variation of the Sine-Gordon theory action about its static kink gives ${\Delta }_x=-{\partial }_x^2+1-2{\mathrm{sech}}^{2}x$. Let $A={\partial }_x+\tanh x$. Then ${\Delta }_x=A^{†}A$ and $A{A}^{†}=-{\partial }_x^2+1$. Thus ${\Delta }_x$ is nonnegative, with its translational zero mode of a sine-Gordon kink and a continuum at ${\omega }^2=k^2+1$. The construction is the supersymmetric factorization of the one-soliton potential.

The normalized eigenfunction ${\psi }_0(x)=\mathrm{sech}x/\sqrt{2}$ has zero eigenvalue under the Sine-Gordon kink fluctuation operator. It is proportional to ${\partial }_x{\varphi }_K$ and comes from shifting the collective coordinate of the kink. Its frequency is zero, so it contributes no oscillator zero-point energy; it must be handled separately from a Gaussian functional determinant.