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New conditional symmetries and exact solutions of nonlinear reaction–diffusion–convection equations

“…To the best of our knowledge many of the solutions presented above are new. However, we noted that some of them can be derived from the recent paper [46]. In fact, if one applies substitution (36) to the RDC equation (72) and its solutions (69)-(71), then equation (54) [46] with (65)-(66) [46] and α(s) = s n , and solutions (69)-(71) [46] are exactly obtained.…”

“…However, we noted that some of them can be derived from the recent paper [46]. In fact, if one applies substitution (36) to the RDC equation (72) and its solutions (69)-(71), then equation (54) [46] with (65)-(66) [46] and α(s) = s n , and solutions (69)-(71) [46] are exactly obtained. Nevertheless, the authors of that paper do not use any symmetries to construct exact solutions, formula (72) [46] is nothing else but the equation Q(V ) = 0, where Q is the conditional symmetry operator (47) of (46).…”

“…In fact, if one applies substitution (36) to the RDC equation (72) and its solutions (69)-(71), then equation (54) [46] with (65)-(66) [46] and α(s) = s n , and solutions (69)-(71) [46] are exactly obtained. Nevertheless, the authors of that paper do not use any symmetries to construct exact solutions, formula (72) [46] is nothing else but the equation Q(V ) = 0, where Q is the conditional symmetry operator (47) of (46). Thus, we obtain new confirmation of the known idea (see, i.e., [12,21,23]) that any exact solution can be obtained by the relevant Lie or conditional symmetry operator.…”

“…Applying substitution (36) to equation (46) and operator (47) we obtain case (i) of the theorem (note one should use new notations…”

“…It turns out that the last equation can be satisfied only under condition h g = const (see the proof below). Setting h g = const into (48) and making the relevant calculations, we only obtain the Lie symmetry operators and a particular case of operator (47) and equation (46). For example, if h g = 0 then the system…”