{c}

If $Q''-Q+Q^p=0$, then
$$
Q_c(x)=c^{1/(p-1)}Q(\sqrt c\,x)
$$
satisfies $Q_c''-cQ_c+Q_c^p=0$, and $u(t,x)=Q_c(x-ct)$ is a traveling-wave solution of the <Generalized Korteweg–De Vries equation>.

Generalized KdV solitary wave

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

{c}
{title2=$u_t+(u_{xx}+u^p)_x=0$}
{wiki=Korteweg–De_Vries_equation}

The generalized Korteweg–De Vries equation is the <nonlinear partial differential equation>
$$
u_t+\partial_x(u_{xx}+u^p)=0.
$$
The cases below the exponent $p=5$ are energy-subcritical in $H^1(\mathbb R)$.

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