On the nonlinear self-adjointness and local conservation laws for a class of evolution equations unifying many models

A family of wave equations with some remarkable properties

“…Later, in [37], two of us have established a new conservation law for = −2. In the present paper we find new conservation laws for = 0 and = −2/3 obtained from the point symmetries of (1) and the results proved in [19,20,12]. This is done in section 3.…”

“…provides a nonlocal conservation law for (1), see [19,20,12] for further details. A natural observation from Lemma 3.1 and the components (14) is that the vector established relies upon the variable v and, therefore, it does not provide a conservation law for equation (1) itself, but to it and its corresponding adjoint.…”

Nonlinear Self-Adjoint Classification of a Burgers-KdV Family of Equations

“…Considering Harry-Dym equation, it is now clear that it is not strictly self-adjoint, but it is quasi selfadjoint. In [15], we proved that adjoint equation (12) to (11) is also equivalent to itself by considering the substitutions:…”

“…Recently [15], we considered a class of time dependent equations up to fifth-order and we obtained necessary and sufficient conditions for determining the nonlinearly selfadjoint subclasses. Nonlinear self-adjointness of equations up to fifth-order can also be found in [16][17][18].…”

Symmetries and Conservation Laws for Some Compacton Equation

“…The concept of strictly self-adjoint equations has been generalized [10][11][12]. After Ibragimov's results, several papers appeared to be concerned with self-adjointness and its applications to PDEs [13][14][15][16][17][18][19][20][21][22][23][24]. This method is a special case of the formula presented in [7].…”