KdV mass and energy conservation

For sufficiently smooth real solutions of the Generalized Korteweg–De Vries equation with enough decay to remove boundary terms, the mass and energyare 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 givesbecause $u_t=-F_x$. Extending these identities to rougher solution classes requires the corresponding approximation and local well-posedness theory; the displayed computation is the smooth-decay argument.