Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 33D / b / Solution

{title2=$u_{z\bar z}=\frac12\sinh(2u)$}
{wiki}

The sinh-Gordon equation is an integrable nonlinear wave equation. Under the one-dimensional reduction $u=u(x)$ with $z=x+iy$, it becomes
$$
u_{xx}=2\sinh(2u).
$$
Writing $\phi=2u$ gives $\phi''=4\sinh\phi$ and the first integral
$$
\frac12(\phi')^2-4\cosh\phi=\text{constant}.
$$


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