the condition (\[qsa0.eq2\]).

\[qsa:exa2\] Let us consider the Fornberg-Whitham equation [@for-whi] $$\label{FW}
 u_t- u_{txx}-u u_{xxx}-3 u_x u_{xx}+u u_x+u_x=0.$$ Eqs. (\[saeq.2\])-(\[saeq.3\]) give the following adjoint equation: $$\label{FW:adj}
 F^* \equiv - v_t + v_{txx} + u v_{xxx} - u v_x - v_x = 0.$$ It is manifest from the equations (\[FW\]) and (\[FW:adj\]) that the Fornberg-Whitham equation is not strictly self-adjoint. Let us test it for quasi self-adjointness. Inserting in (\[FW:adj\]) the substitution $v = \varphi(u)$ and its derivatives $$v_t = \varphi' u_t, \quad  v_x = \varphi' u_x, \quad  v_{xx} = \varphi' u_{xx}
 + \varphi'' u_x^2, \quad  v_{tx} = \varphi' u_{tx}
 + \varphi'' u_t u_x, \ldots\,,$$ then writing the condition (\[qsa.2\]) and comparing the coefficients for $u_t, \ u_{tx}, \ u_{xx}, \ldots$ one can verify that $\varphi'(u) = 0.$ Hence, Eq. (\[FW\]) is quasi self-adjoint but does not satisfy the condition (\[qsa0.eq2\]).

Strict self-adjointness via multipliers {#samul}
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It is commonly known that numerous linear equations used in practice, e.g. linear evolution equations, are not self-adjoint in the classical meaning of the self-adjointness. Likewise, useful nonlinear equations such as the nonlinear heat equation, the Burgers equation, etc. are not strictly self-adjoint. We will see here that these and many other equations can 