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Use $\hslash =c=1$ in the field-theory calculations. Work with signature $(+,-)$ and $a\ne 0$. Since $x=(x_{+}+x_{-})$ and $t=(x_{+}-x_{-})$, ${\partial }_{+}={\partial }_x+{\partial }_t$, ${\partial }_{-}={\partial }_x-{\partial }_t$, and the Sine-Gordon equation is ${\varphi }_{+-}=\sin \varphi$. Introduce $w=(\rho +\kappa )/2$ and $d=(\rho -\kappa )/2$. The specified Sine-Gordon Bäcklund transformation becomes $d_{+}=a\sin w$, $w_{-}=a^{-1}\sin d$. Its compatibility givesAdding and subtracting these equations establishesThus a compatible transformed field satisfies the same Sine-Gordon equation. Smoothness is needed to interchange mixed derivatives. The statement cannot literally include $a=0$, since one defining equation contains $1/a$.

Generating the kink. With zero seed, the two first-order equations give ${\rho }_{+}=2a\sin (\rho /2)$ and ${\rho }_{-}=2a^{-1}\sin (\rho /2)$. On a nonconstant branch put $p=\tan (\rho /4)$; then $p_{+}=ap$, $p_{-}=a^{-1}p$. Integration givesFor $a>0,C>0$, this is the Sine-Gordon kink, after absorbing $C$ in the center position. Its speed and Lorentz factor obeyChanging the sign of $a$ or of the exponential coefficient selects the corresponding antikink orientation; $|v|<1$ whenever $a$ is finite and nonzero. Constant vacuum branches, omitted by division by $\sin (\rho /2)$, can be added separately.

The continuously parametrized Bäcklund transformation does more than produce this one solution. Successive compatible transformations and Bianchi permutability for sine-Gordon Bäcklund transformations construct multisoliton solutions; expansion of the generating relations gives the local conserved-charge hierarchy of sine-Gordon theory. Together with its Lax pair and inverse scattering transform, this is the structure behind classical integrability, elastic scattering and the absence of generic radiative energy loss in soliton collisions. Existence of a transformation alone is not a proof that every generated conserved charge is independent and in involution; those are additional properties of the integrable system.

Transforming the static kink at $a=1$. The seed obeys ${\kappa }_x=2\mathrm{sech}x$, $\sin (\kappa /2)=\mathrm{sech}x$, $\cos (\kappa /2)=-\tanh x$. Adding and subtracting the two Sine-Gordon Bäcklund transformation equations yieldsAgain set $p=\tan (\rho /4)$. These simplify to $p_x=-p\tanh x$, $p_t=\mathrm{sech}x$. The first gives $p=f(t)/\cosh x$, and the second gives $f^{\prime }=1$. ThereforeContinuous inverse-tangent branches may be chosen without creating artificial jumps. At spatial infinity the field approaches the same vacuum, so its net topological charge is zero. For large $|t-t_0|$ there are two well-separated transitions near $x_{\pm }=\pm \mathrm{arcosh}|t-t_0|$, with speeds tending to zero. The two transitions exchange their kink/antikink orientations as they pass through the collision, while the field remains smooth.

This is the Sine-Gordon threshold kink-antikink solution, a separatrix between finite-speed scattering and the bounded Sine-Gordon breather motion, rather than a finite-period breather. In fact it is the $v\to 0$ limit of the scattering field in question 3. At $t=t_0$, $\rho =0$ and ${\rho }_t=4\mathrm{sech}x$, so the reduced energy is $\int {\rho }_t^{2}dx/2=16$, exactly twice the reduced single-kink energy $8$.

For mass comparisons below, the original PDF really prints $m^2/\beta$. Restoring physical coordinates $X=x/m$, $T=t/m$ gives the angular-field action ${\beta }^{-1}\int dtdx[\frac{1}{2}(\partial \varphi {)}^2+\cos \varphi -1]$ and literal kink mass $8m/\beta$, for $\beta >0$. The usual Sine-Gordon theory instead has prefactor $m^2/{\beta }^2$ and mass $8m/{\beta }^2$. The equation of motion and every field above are unchanged by this overall normalization; the quantum scattering comparison in question 3 is not.

Use the Minkowski metric with signature $(+,-)$, so the Sine-Gordon equation is ${\varphi }_{tt}-{\varphi }_{xx}+\sin \varphi =0$. The light-cone coordinates here satisfyIn particular, there is no extra factor of four with this coordinate normalization. The Bäcklund transformation requires $a\ne 0$, because one of its equations contains $a^{-1}$.

Put $F=(\psi +\varphi )/2$ and $G=(\psi -\varphi )/2$. In the parameter convention of this paper the Sine-Gordon Bäcklund transformation gives $G_{+}=a\sin F$ and $F_{-}=a^{-1}\sin G$. Differentiating, for a twice differentiable transformed field, givesSince $\psi =F+G$, the addition formula yields ${\psi }_{+-}=\sin (F+G)=\sin \psi$. Therefore the transformed field satisfies the same Sine-Gordon equation:Subtracting the two differentiated equations also gives ${\varphi }_{+-}=\sin \varphi$, showing the compatibility with the seed equation. This proves the unheaded request before the numbered parts, without assuming a particular soliton form.

Compatibility of the auxiliary system, computed on a fundamental matrix of solutions, gives the zero-curvature conditionIndeed equality of the mixed derivatives gives $(U_t+UV-V_x-VU)\Psi =0$. A single possibly zero vector solution would not justify the matrix identity; compatibility means a full fundamental family.

Put $H=W_t-VW$. Using the zero-curvature condition gives $H_x=UH$. At $x=0$, $W_t=0$ and hence $H(0)=-V(0)$. Uniqueness of the initial-value problem gives $H=-WV(0)$, or $W_t=VW-WV(0)$. Periodicity implies $V(2\pi )=V(0)=v$, so the monodromy matrix $w=W(2\pi )$ obeys the Lax equationCyclicity of the trace yields $\frac{d}{dt}\mathrm{tr}(w^k)=k\mathrm{tr}(w^{k-1}[v,w])=0$ for every $k\ge 1$.

There is a genuine factor-of-two error in the printed sine-Gordon pair. Assign the second printed matrix to $U$ and the first to $V_{\mathrm{print}}$:Direct multiplication givesThus the printed matrices encode $u_{xt}=2\sin u$.

For the stated Sine-Gordon equation, explicitly repair the first prefactor by taking $V=V_{\mathrm{print}}/2$, that is $1/(4i\lambda )$. Its zero-curvature condition becomes $u_{xt}=\sin u$. For this corrected pair, $\mathrm{tr}(w(t,\lambda {)}^k)$ is conserved for every $\lambda \ne 0$. Since $U$ is entire in $\lambda$, so are $W$ and these traces. Their Taylor coefficients in $\lambda$ supply an infinite family of first integrals, by the identity theorem extending conservation through $\lambda =0$. At a fixed spectral parameter the higher traces of a $2\times 2$ matrix are related by the Cayley-Hamilton theorem; no independence of the resulting first integrals is claimed.

The transformationsform a one-parameter group with infinitesimal generatorUnder this scaling, ${\partial }_X$ has weight $-1$ and ${\partial }_T$ has weight $1$, so $u_{XT}$ has weight zero, as does $\sin u$. Thus $u_{XT}=\sin u$ is invariant and $g^s$ is a Scaling symmetry of the sine-Gordon equation.

The productis invariant under the group. A group-invariant solution therefore has the form $u(X,T)=F(z)$. Sincethe Sine-Gordon equation reduces to

Now put $w=e^{iF}$. ThenSubstitution and multiplication by $iw$ giveHenceThis is the Painlevé III equation with parameters $(\alpha ,\beta ,\gamma ,\delta )=(1/2,-1/2,0,0)$.

For real $f\in L^2({\mathbb{R}}^3)$, this energy functional on $H^1({\mathbb{R}}^3)$ is coercive and attains its minimum by the direct method in the calculus of variations. The nonnegative potential term is weakly lower semicontinuous by local Sobolev compactness gives lower semicontinuity of a nonnegative integral. Its Euler-Lagrange equation is $-\Delta u+u+\sin u=f$ in the weak solution sense. The extra linear restoring term screens the Sine-Gordon equation nonlinearity. The scalar potential $V(s)=s^2/2+1-\cos s$ is convex because $V^{\prime \prime }(s)=1+\cos s\ge 0$; hence any weak critical point is a minimizer. The elliptic regularity estimate for $-\Delta +1$, together with $|\sin u|\le |u|$, gives $u\in H^2$. The Sobolev inequality then puts $u$ in $W^{1,6}$, so Morrey's inequality and uniformly continuous integrable functions vanish at infinity give a continuous representative tending to zero.

A sine-Gordon breather is a spatially localized solution that is periodic in time and can be interpreted through a complex-conjugate pair of discrete scattering eigenvalues.

For real parameters with ${\kappa }_i^2-b_i^2=1$, set $E_i=e^{{\kappa }_{i}x-b_{i}t+{\gamma }_i}$ and $a_{ij}=[({\kappa }_i-{\kappa }_j{)}^2-(b_i-b_j{)}^2]/[({\kappa }_i+{\kappa }_j{)}^2-(b_i+b_j{)}^2]$. Sum ${\prod }_i{E}_i^{{\mu }_i}{\prod }_{i<j}{a}_{ij}^{{\mu }_i{\mu }_j}$ over binary vectors of even parity for $f$ and odd parity for $g$. The Sine-Gordon equation solution is the continuous field $4\arg (f+ig)$. In the all-kink sector, ${\kappa }_i=\cosh {\theta }_i$, $b_i=\sinh {\theta }_i$ and $a_{ij}=-{\tanh }^2[({\theta }_i-{\theta }_j)/2]$. Treating these coefficients as positive would change the solution. Distinct rapidities give separated incoming and outgoing solitons.

A relativistic real scalar field theory with a periodic cosine potential. In physical coordinates $y^{\mu }$, one normalization is $\mathcal{L}=\frac{1}{2}{\partial }_{\mu }\phi {\partial }^{\mu }\phi -m^2(1-\cos \beta \phi )/{\beta }^2$. With $\varphi =\beta \phi$ and $x^{\mu }=m{y}^{\mu }$, its action is $S={\beta }^{-2}\int d^{2}x[\frac{1}{2}(\partial \varphi {)}^2+\cos \varphi -1]$. Its Euler-Lagrange field equation is the Sine-Gordon equation. Distinct scalar-field vacua differ by $2\pi$ in $\varphi$, permitting a Sine-Gordon kink.