in Eq. (\[Burg:mult.eq13\]) we see that *Eq.* (\[Burg:mult.eq14\]) *becomes strictly self-adjoint if we rewrite it in the form* $$\label{Burg:mult.eq15}
 \frac{u_t}{u^3} = \frac{u_{xx}}{u}\,\cdot$$

Harry Dym equation {#samul:6}
------------------

Taking in Example \[Burg:mult.exa4\] from Section \[samul:1\] $n = 3$ and $f(u) = u^3$ we see that the Harry Dym equation $$\label{Burg:mult.eq16}
 u_t - u^3 u_{xxx} = 0$$ becomes strictly elf-adjoint upon rewriting it in the form $$\frac{u_t}{u^4} - \frac{u_{xxx}}{u} = 0.$$

Kompaneets equation {#samul:7}
-------------------

The equations considered in Sections \[samul:1\] - \[samul:6\] are quasi self-adjoint. For example, for Eq. (\[Burg:mult.eq12\]) we have $$F^* = - v_t - D_x^n(f(u) v) -  v f'(u)  u_n,$$ whence making the substitution $$v = \frac{1}{f(u)}$$ we obtain $$F^* = \frac{f'}{f^2}\, u_t  -  \frac{ f'}{f}\,  u_n
= \frac{f'}{f^2} [u_t  - f(u)\,  u_n].$$ Hence, Eq. (\[Burg:mult.eq12\]) is quasi self-adjoint.

\[Burg:mult.exa7\] The Kompaneets equation $$\label{Burg:mult.Komp1}
 u_t = \frac{1}{x^2}\, D_x\left[x^4 (u_x + u + u^2)\right]$$ provides an example of an equation that is not quasi self-adjoint. Indeed, Eq. (\[Burg:mult.Komp1\]) has the formal Lagrangian $${\cal L} = v [- u_t + x^2 u_{xx} + (x^2 + 4 x + 2 x^2 u) u_x
 + 4 x (u + u^2)].$$ The calculation yields the following adjoint equation to (\[Burg:mult.Komp1\]): $$\label{Burg:mult.Komp2}
 \frac{\delta {\cal 