= KdV mass and energy conservation
{c}
= KdV mass conservation
{c}
{synonym}
= KdV energy conservation
{c}
{synonym}
For sufficiently smooth real solutions of the <Generalized Korteweg–De Vries equation> with enough decay to remove boundary terms, the <mass> and <energy>
$$
M(u)=\int_{\mathbb R}u^2\,dx,\qquad
E(u)=\frac12\int_{\mathbb R}u_x^2\,dx-\frac1{p+1}\int_{\mathbb R}u^{p+1}\,dx
$$
are constant in time. To check <mass conservation>, multiply the equation by $2u$ and use <integration by parts>. The linear term gives $\int u u_{xxx}=0$, and the nonlinear term is a total <derivative> proportional to $(u^{p+1})_x$.
For <energy conservation>, put $F=u_{xx}+u^p$. Differentiating the <energy functional> gives
$$
\frac{dE}{dt}=-\int F u_t\,dx=\int F F_x\,dx=0,
$$
because $u_t=-F_x$. Extending these identities to rougher solution classes requires the corresponding approximation and local well-posedness theory; the displayed computation is the smooth-decay argument.
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