Differentiating Ioana's $(A)$ and equating it with his(her?) $(B)$, we have the symbolic system

$$\left( \begin{array}{cc|c}

1 & -1 & 2x\\

1+x& 1-x&3x^{2}\\

\end{array} \right)

\implies

\left( \begin{array}{c}

\varphi'(X)\\

\psi'(Y)

\end{array} \right)

=

\left( \begin{array}{c}

X\\

-Y

\end{array} \right)

\text{ where X, Y are the arguments of $\varphi, \psi$ resp. }

\implies

\left( \begin{array}{c}

\varphi(X)\\

\psi(Y)\end{array} \right)

=

\left( \begin{array}{c}

X^{2}/2+C_{1}\\

-Y^{2}/2+C_{2}

\end{array} \right)

\implies

u=(x+t)^{2}/2 - (x-t)^{2}/2 + const.

$$

Fixed now. For uniqueness we impose that the characteristics intersect the initial data, that is, precisely when

$$x^{2}-2x+2C, x^{2}+2x-2C$$

both have solution. This happens whenever

$$-t-1/2\leq x\leq t+1/2.$$