# How to find the conserved quantities of the Kirchhoff equation?

Consider the Kirchhoff equation, given by $$u_{tt}-\left(1+\int_{\mathbb{R}} u_x^2\;dx\right)u_{xx}+f(u)=0, (x,t) \in \mathbb{R}\times \mathbb{R}_+$$ where $$f(u)=u-u^{2r+1}$$, for $$r \in \mathbb{N}$$. How to find the conserved quantitie of this equation?

• Multiply by $$u_x$$ and integrate, you get $$\int u_{tt} u_x ~dx = 0$$ so $$\partial_t \int u_{t} u_x ~dx - \int u_t u_{tx} ~dx = 0$$ the second term integrates to zero.
• Multiply by $$u_t$$ and integrate by parts you get $$\int u_{tt} u_t + (1 + \int u_x^2 ~dx) u_{xt} u_x + uu_t - u^{2r+1} u_t ~dx = 0$$ This you can rewrite as $$\partial_t \int \frac12 u_t^2 + \frac12 (1 + \frac12 \int u_x^2~dx) u_x^2 + \frac12 u^2 - \frac1{2r+2} u^{2r+2} ~dx = 0$$
• Why multiplying by $u_x$ and integrante we get only $\int u_{tt}u_x \; dx=0$? And the rest of the terms? Commented Jun 10, 2020 at 13:23
• Integration of a total derivative yields 0. $$\int u_{xx} u_x = \int \partial_x (u_x)^2 = 0$$ and similarly the $f(u)$ terms. $$\int f(u) u_x = \int \partial_x F(u) = 0$$ where $F$ is the primitive to $f$. Commented Jun 10, 2020 at 17:17