Conservation laws - Airy's equation

Question

I'm considering Airy equation $u_t+u_{xxx}=0$ and want to derive two conservation laws for a solution $u(t; x)$ $$\frac{d}{dt}\int{u}dx=0$$ and $$\frac{d}{dt}\int{u^2}dx=0$$ Using Fourier transform I have derive the solution formula, but stuck here.. Can someone help me?

Answer 1

$\newcommand{\+}{^{\dagger}} \newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle} \newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack} \newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,} \newcommand{\dd}{{\rm d}} \newcommand{\down}{\downarrow} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,{\rm e}^{#1}\,} \newcommand{\fermi}{\,{\rm f}} \newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,} \newcommand{\half}{{1 \over 2}} \newcommand{\ic}{{\rm i}} \newcommand{\iff}{\Longleftrightarrow} \newcommand{\imp}{\Longrightarrow} \newcommand{\isdiv}{\,\left.\right\vert\,} \newcommand{\ket}[1]{\left\vert #1\right\rangle} \newcommand{\ol}[1]{\overline{#1}} \newcommand{\pars}[1]{\left(\, #1 \,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\pp}{{\cal P}} \newcommand{\root}[2][]{\,\sqrt[#1]{\vphantom{\large A}\,#2\,}\,} \newcommand{\sech}{\,{\rm sech}} \newcommand{\sgn}{\,{\rm sgn}} \newcommand{\totald}[3][]{\frac{{\rm d}^{#1} #2}{{\rm d} #3^{#1}}} \newcommand{\ul}[1]{\underline{#1}} \newcommand{\verts}[1]{\left\vert\, #1 \,\right\vert} \newcommand{\wt}[1]{\widetilde{#1}} \newcommand{\uu}{\,{\rm u}}$ \begin{align} \totald{}{t}\int_{a}^{b}\uu^{2}\,\dd x&=\int_{a}^{b}2\uu\uu_{t}\,\dd x =-2\int_{a}^{b}\uu\uu_{xxx}\,\dd x =\left.\vphantom{\Large A}-2\uu\uu_{xx}\right\vert_{a}^{b} + 2\int_{a}^{b}\uu_{xx}\uu_{x}\,\dd x \\[3mm]&=\left.\vphantom{\Large A}-2\uu\uu_{xx}\right\vert_{a}^{b} +\int_{a}^{b}\partiald{\uu_{x}^{2}}{x}\,\dd x =\bracks{-2\uu\uu_{xx} + \uu_{x}^{2}}_{a}^{b} \end{align}

$$ \bracks{2\uu \uu_{xx} - \uu_{x}^{2}}_{a}^{b} +\totald{}{t}\int_{a}^{b}\uu^{2}\,\dd x = 0 $$

Whenever $\ds{\bracks{2\uu\uu_{xx} - \uu_{x}^{2}}_{a}^{b} = 0}$ we'll have $\ds{\int_{a}^{b}\uu^{2}\pars{x,t}\,\dd x}$ independent of $\ds{t}$.