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An integrable partial differential equation has enough exact structure, such as a Lax pair, infinitely many conserved quantities, or an inverse scattering transform, to reduce its evolution to auxiliary linear or finite-dimensional data.

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.

The Korteweg-De Vries equation is the compatibility condition $L_t=[P,L]$ for the Lax pairFor rapidly decaying initial data, first solve the direct KdV Schrodinger spectral problemComparing its Jost solutions at the two spatial infinities produces the KdV scattering data: the reflection coefficient $R(k)$ on the continuous spectrum, discrete eigenvalues $-{\kappa }_n^2$, and norming amplitudes $c_n$ for the bound states.

The Lax evolution is isospectral. More precisely, the Time evolution of KdV scattering data isThis is the simple linear evolution that makes the nonlinear initial-value problem tractable.

To invert the data, formSolve the Gelfand-Levitan-Marchenko equationand reconstructDirect scattering at $t=0$, linear evolution of the data, and this inverse step constitute the inverse scattering transform.

Putand seek the rank-one Gelfand-Levitan-Marchenko kernel in the separable formSubstitution into the Gelfand-Levitan-Marchenko equation givessoand hence

WriteThe reconstruction formula for this inverse scattering transform convention yieldsIf $q=e^{-2s}$, then $4q/(1+q{)}^2={\mathrm{sech}}^{2}s$. Takingtherefore produces the required one-soliton of the Korteweg-De Vries equation, withHere the usual bound-state norming constant has ${\beta }_0>0$ and $\chi >0$, so $\varphi$ is real.

In light-cone coordinates, the sine-Gordon equation can be written $u_{XT}=\sin u$. Its inverse scattering transform, solitons, breathers, and scaling symmetries make it an integrable system.