Let $D_\mu=\partial_\mu+A_\mu$. A <Lax pair> with spectral parameter $\lambda$ is
$$
\boxed{L(\lambda)=D_w-\lambda D_{\bar z},
\qquad
M(\lambda)=D_z+\lambda D_{\bar w}}.
$$
The coefficients of $[L,M]=0$ at orders $1,\lambda,\lambda^2$ are respectively
$$
F_{wz}=0,\qquad F_{w\bar w}+F_{z\bar z}=0,\qquad F_{\bar w\bar z}=0,
$$
which are precisely the anti-self-dual Yang-Mills equations.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 313 / 2 / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-313/2/ii/solution