= Solution
With zero seed, both <Bäcklund transformation> equations concern $\phi_a/2$. On a nonconstant branch, separation of variables uses $d\log|\tan(\phi_a/4)|=d\phi_a/[2\sin(\phi_a/2)]$, giving
$$
\tan\frac{\phi_a}{4}=C\exp(a x_++a^{-1}x_-),\qquad a\ne0.
$$
For $C>0$, absorb its magnitude into an additive constant $c=\log C$ in the exponent. A convenient smooth representative is
$$
\boxed{\phi_a(x,t)=4\arctan\exp\left[\frac{a+a^{-1}}2x+\frac{a-a^{-1}}2t+c\right].}
$$
The constant-phase condition for this traveling profile determines its <velocity>:
$$
\boxed{v_a=\frac{1-a^2}{1+a^2},\qquad\gamma_a=\frac{|a+a^{-1}|}{2}=\frac1{\sqrt{1-v_a^2}}.}
$$
\b[The profile is a traveling <soliton> with <velocity> $v_a=(1-a^2)/(1+a^2)$, strictly between $-1$ and $1$.] The <scalar-field vacua> on the two sides differ by $2\pi$. With the <topological charge> convention $Q=[\phi(+\infty)-\phi(-\infty)]/(2\pi)$, this branch has $Q=\operatorname{sgn}a$: it is a <Sine-Gordon kink> for $a>0$ and an <antikink> for $a<0$. Its width is proportional to $\gamma_a^{-1}$, and its derivative decays exponentially away from its center, giving a <finite-energy field configuration>.
For completeness, the classical rest <mass> in this paper's coupling convention is $8m/\beta$, as derived in Question 2; a <Lorentz boost> gives energy $8m\gamma_a/\beta$. This localized, topologically protected traveling field is the required <classical field-theory soliton>. A negative $C$ reverses the field and hence the <topological charge>; $C=0$ gives the vacuum rather than a <soliton>. Vacuum shifts by $4\pi$ can be made without changing the displayed <Bäcklund transformation> equations.
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