x u - 1) \varphi(u) = 0$ and yields $\varphi(u) = 0.$ Hence the Kompaneets equation is not quasi self-adjoint because the condition (\[qsa.eq3\]) is not satisfied.

But we can rewrite Eq. (\[Burg:mult.Komp1\]) in the strictly self-adjoint form by using a more general multiplier than above, namely, the multiplier $$\label{Burg:mult.Komp3}
 \mu = \frac{x^2}{u}\,\cdot$$ Indeed, upon multiplying by this $\mu$  Eq. (\[Burg:mult.Komp1\]) is written $$\frac{x^2}{u}\,u_t = \frac{1}{u}\,D_x\left[x^4 (u_x + u +
 u^2)\right].$$ Its formal Lagrangian $${\cal L} = \frac{v}{u}\,\left\{- x^2 u_t + D_x\left[x^4 (u_x + u +
u^2)\right]\right\}$$ satisfies the strict self-adjointness condition (\[saeq.5\]) with $\lambda = - 1:$ $$\frac{\delta {\cal L}}{\delta u}\bigg|_{v = u} = - \frac{1}{u}\,\left\{- x^2 u_t + D_x\left[x^4 (u_x + u +
u^2)\right]\right\}.$$

\[Burg:mult.Komp2.rem\] Note that $v = x^2$ solves Eq. (\[Burg:mult.Komp2\]) for any $u.$ The connection of this solution with the multiplier (\[Burg:mult.Komp3\]) is discussed in Section \[gsa:6\]. See also Section \[gsa:7\].

General concept of nonlinear self-adjointness
=============================================

\[gsa\] Motivated by the examples discussed in Sections \[sa:eveq.int\] and \[samul\] as well as other similar examples, I suggest here the general concept of *nonlinear self-adjointness* of systems consisting of any number of equations with $m$ dependent variables. This concept encapsulates Definition \[saeq:def\] of strict self-adjointness and Definition 