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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2021

Paper 154

Differentiate the proposed energy and integrate the kinetic term by parts:
$$
\frac d{dt}E(u(t))
=\int_{\mathbb R}(-u_{xx}-u^p)u_t\,dx.
$$
Putting $F=u_{xx}+u^p$, the equation says $u_t=-F_x$, so
$$
\frac d{dt}E(u(t))
=\int_{\mathbb R}F F_x\,dx
=\frac12\int_{\mathbb R}\partial_x(F^2)\,dx=0.
$$
Hence <KdV energy conservation> gives
$$
\boxed{E(u(t))=E(u_0)}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 154 / 1 / 2 / Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 154 / 1 / 2 / Solution

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Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 154 / 1 / 2 / Solution

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