equation to the heat equation (\[sa:eveq.exa1:eq1\]) is written $- v_t - v_{xx}=
0,$ or $$\label{sa:eveq.exa1:eq4}
 v_t + v_{xx}= 0.$$

The derivation of the adjoint equation (\[sa:eveq.exa1:eq4\]) and the adjoint operator (\[sa:eveq.exa1:eq3\]) by the definition (\[saeq.3\]) is much simpler. Indeed, we have: $$F^* = \frac{\delta (v u_t - v u_{xx})}{\delta u} =
   - D_t(v) - D_x^2(v) = - (v_t + v_{xx}).$$

Self-adjointness and quasi self-adjointness {#sa:eveq.4}
-------------------------------------------

Recall that a linear differential operator $F$ is called a *self-adjoint operator* if it is identical with its adjoint operator, $F = F^*.$ Then the equation $F[u] = 0$ is also said to be self-adjoint. Thus, the self-adjointness of a *linear differential equation* $F[u] = 0$ means that the adjoint equation $F^*[v] = 0$ coincides with $F[u] = 0$ upon the substitution $v = u.$ This property has been extended to nonlinear equations in [@ibr06a]. It will be called here the *strict self-adjointness* and defined as follows.

\[saeq:def\] We say that the differential equation (\[saeq.1\]) is *strictly self-adjoint* if the adjoint equation (\[saeq.2\]) becomes equivalent to the original equation (\[saeq.1\]) upon the substitution $$\label{saeq.4}
 v = u.$$ It means that the equation $$\label{saeq.5}
 F^*\big(x, u, u, \ldots, u_{(s)}, u_{(s)}\big) = \lambda\, F\big(x, u,
 \ldots, 