Let $\zeta\in\mathbb{CP}^1$ be the <spectral parameter>. Define the two covariant differential operators
$$
L(\zeta)=D_1+iD_2-\zeta(D_3-iD_\tau),
$$
$$
M(\zeta)=D_3+iD_\tau+\zeta(D_1-iD_2).
$$
The auxiliary system
$$
L(\zeta)\Psi=0,
\qquad
M(\zeta)\Psi=0
$$
is compatible exactly when $[L(\zeta),M(\zeta)]=0$ for every $\zeta$. The constant and quadratic coefficients give
$$
F_{13}-F_{2\tau}=0,
\qquad
F_{1\tau}+F_{23}=0,
$$
and their equivalent conjugate equations, while the linear coefficient gives
$$
F_{12}+F_{3\tau}=0.
$$
Since $F_{i\tau}=-F_{\tau i}$, these are precisely
$$
F_{\tau1}=F_{23},
\qquad F_{\tau2}=F_{31},
\qquad F_{\tau3}=F_{12}.
$$
Thus
$$
\boxed{(L(\zeta),M(\zeta))}
$$
is a <Lax pair for the anti-self-dual Yang-Mills equations>. In temporal gauge one simply sets $D_\tau=\partial_\tau$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 313 / 3 / d / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-313/3/d/solution