these, say $M_n$, exceeds the sum of the other two, then $c_n=1$ and the two other coefficients vanish. Thus the weak limit produces in this case exactly the branch $u_n^B(x,t)$ through multiphase averaging[^1]. On the other hand, if none of the $M_n$ exceeds the sum of the other two, then $M_0$, $M_1$, and $M_2$ are the side lengths of a triangle, and the weak limit is a genuine weighted average of the three branches, with weights proportional to the opposite angles: $$\begin{split}
c_0&=\frac{1}{\pi}\arccos\left(\frac{M_1^2+M_2^2-M_0^2}{2M_1M_2}\right)\\
c_1&=\frac{1}{\pi}\arccos\left(\frac{M_0^2+M_2^2-M_1^2}{2M_0M_2}\right)\\
c_2&=\frac{1}{\pi}\arccos\left(\frac{M_0^2+M_1^2-M_2^2}{2M_0M_1}\right).
\end{split}$$ The most significant aspect of this analysis is that the weak limit depends on information other than just the initial condition $u_0$ for Burgers’ equation since the functions $M_n(x,t)$ involve also the initial wave function amplitude $A$. This makes the evaluation of the weak limit a more complicated procedure than in the case of the BO equation.

A further connection between the BO equation and the linear Schrödinger equation in the zero-dispersion limit will be made in §\[sec:reflectionless\].

Relevant Aspects of the Inverse Scattering Transform for the BO Cauchy Problem {#sec:background}
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The Lax pair for the BO equation and its basic properties
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The Lax pair [@Bock; @1979], whose compatibility condition is the 