Write the <Generalized Korteweg–De Vries equation> as
$$
u_t=-\partial_x(u_{xx}+u^p).
$$
For a sufficiently regular solution, <integration by parts> gives
$$
\frac d{dt}\int_{\mathbb R}u^2\,dx
=-2\int_{\mathbb R}u\,\partial_x(u_{xx}+u^p)\,dx
=2\int_{\mathbb R}u_xu_{xx}\,dx
+2\int_{\mathbb R}u^pu_x\,dx=0.
$$
Thus <KdV mass conservation> gives
$$
\boxed{\|u(t)\|_2=\|u_0\|_2}
$$
is conserved.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-154/1/1/solution