u_x^2\right] = 0.$$ Indeed, the formal Lagrangian (\[sa:Lag\]) for Eq. (\[Burg:mult.eq11\]) is written $${\cal L} = \frac{v}{u}\,\left[u_t - k(u) u_{xx} - k'(u) u_x^2\right].$$ Substituting it in (\[saeq.3\]) we obtain $$\begin{aligned}
 F^* & = - D_t\left(\frac{v}{u}\right) - D_x^2 \left(\frac{v}{u}\,k(u) \right)
 + 2 D_x \left(\frac{v}{u}\,k'(u) u_x \right)\notag\\[1ex]
 & - \frac{v}{u}\,k'(u) u_{xx}
- \frac{v}{u}\,k''(u) u_x^2 - \frac{v}{u^2}\,\left[u_t - k(u) u_{xx}
- k'(u) u_x^2\right].\notag
 \end{aligned}$$ Letting here  $v = u$ we have: $$F^* = - \frac{1}{u}\,\left[u_t - k(u) u_{xx} - k'(u) u_x^2\right].$$ Hence, Eq. (\[Burg:mult.eq9\]) satisfies the strict self-adjointness condition (\[saeq.5\]) with $\lambda = - 1.$

The Burgers equation {#samul:4}
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Taking in (\[Burg:mult.eq5\]) $f (u) = u$ we obtain the strictly self-adjoint form $$\label{Burg:mult.eq6}
 \frac{1}{u}\,[u_t - u_{xx}] -  u_x = 0$$ of the Burgers equation $u_t = u_{xx} +  u u_x.$

Heat conduction in solid hydrogen {#samul:5}
---------------------------------

According to [@ros79], the heat conduction in solid crystalline molecular hydrogen at low pressures is governed by the nonlinear equation (up-to positive constant coefficient) $$\label{hydrogen.eq1}
 u_t =  u^2 \Delta u.$$ It is derived from the Fourier equation $$\rho\,c_* \frac{\partial T}{\partial t}
 = \nabla\cdot (k\,\nabla T)$$ using the empirical information that the density $\rho$ at low pressures has a constant value, whereas the specific heat $c_*$ 