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.

The identically conserved topological current $j$ couples to an external electromagnetic gauge potential through $-A_{\mu }{j}^{\mu }=-A_0{\theta }_x+A_1{\theta }_t$. A gauge transformation changes this coupling only by a divergence. The Euler-Lagrange field equation becomes ${\theta }_{tt}-{\theta }_{xx}+\sin \theta +E=0$, with $E={\partial }_t{A}_1-{\partial }_x{A}_0$. A unit Sine-Gordon kink has unnormalized charge $Q=2\pi$ and mass $8$. For spatially uniform forcing, its field momentum balance is $\dot{P}=QE$ when the boundary stress contributions cancel.

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.

The zero-relative-speed limit of the Sine-Gordon kink-antikink scattering solution is a separatrix at twice the rest-kink energy. A Sine-Gordon Bäcklund transformation of the static kink with equal parameter gives $p_x=-p\tanh x$, $p_t=\mathrm{sech}x$ for $p=\tan (\varphi /4)$, hence the displayed solution. The two asymptotic transitions separate logarithmically in time and their speeds tend to zero; this is not a finite-period Sine-Gordon breather.

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.

For two equal-charge Sine-Gordon kinks with speeds $\pm v$, $0<v<1$, the positive spatial transition satisfies $X_{+}(T)=v|T|-\log v/(m\gamma )+o(1)$, where $\gamma =(1-v^2{)}^{-1/2}$. Labeling a soliton by its preserved rapidity, the right-moving incoming and outgoing intercepts differ by $-2\log v/(m\gamma )$. The soliton time delay at a fixed distant location is minus this shift divided by $v$, giving the displayed negative value. Reversing the entire field gives two antikinks with the same shifts. The sign means an advance relative to extrapolation of the incoming line; labels tied to left and right positions instead exchange velocities during reflection.

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.