How to show the simple equivalence?

Question

I come across the following equivalence of integrations:

$$\int\left[-I \left(h\right){\partial h \over \partial \tau}+{\partial h \over \partial x}{\partial \over \partial \tau} \left({\partial h \over \partial x}\right)\right]\mathrm dx=-\int h^3 \left[{\partial \over \partial x} \left(I(h)+ {\partial^2 h \over \partial x^2} \right) \right]^2\mathrm dx$$

where $h(x,\tau)$ subjects to periodic boundary conditions

$$h[0,\tau]=h[l,\tau]$$ $$h_x[0,\tau]=h_x[l,\tau]$$ $$h_{xx}[0,\tau]=h_{xx}[l,\tau]$$ $$h_{xxx}[0,\tau]=h_{xxx}[l,\tau]$$

But I can't derive it from left-hand side to the right-hand side. I have tried integrate by parts with these periodic B.C.s. Can anyone give me some advice?

Answer 1

What you say does not seem to be correct. Think about function h that only depends on $\tau$:

$$h=h(\tau)$$

Then both sides would be different. This equation is wrong for general case of h.