Singular reduction modules of differential equations

Abstract

The notion of singular reduction modules, i.e., of singular modules of nonclassical (conditional) symmetry, of differential equations is introduced. It is shown that the derivation of nonclassical symmetries for differential equations can be improved by an in-depth prior study of the associated singular modules of vector fields. The form of differential functions and differential equations possessing parameterized families of singular modules is described up to point transformations. Singular cases of finding reduction modules are related to lowering the order of the corresponding reduced equations. As examples, singular reduction modules of evolution equations and secondorder quasi-linear equations are studied. Reductions of differential equations to algebraic equations and to first-order ordinary differential equations are considered in detail within the framework proposed and are related to previous no-go results on nonclassical symmetries.

Classification of reduction operators and exact solutions of variable coefficient Newell–Whitehead–Segel equations

“…Admissible transformations of the class (6) can be easily derived from Theorem 1, where we set a 2 = a 2 = 1 andb = b = 0. It guarantees the complete result since superclass (1) of class (6), is normalized. The result is summarized in the following statement.…”

“…which generate one-parameter Lie groups of point symmetry transformations for equations from class (6). Here we require that…”

“…It was proved therein that the direct approach of reduction, taken in its full generality, is equivalent to the non-classical (conditional symmetry) approach. The enhanced proof can be found in [6].…”

“…The nonclassical reduction operators can be of regular and singular types. The problem of finding singular reduction operators reduces to solving an initial PDE, therefore this case is called the "no-go" case and often omitted in consideration (see more about "no-go" case in [6,18,27,43,44,63]). But even in the case of regular nonclassical reduction operators the problem of their classification for classes of PDEs is difficult.…”

Extended symmetry analysis of generalized Burgers equations

“…In view of results of Section 3, we could postulate the form (10) for the components of Lie symmetry vector fields of equations from the class (1) from the very beginning. Indeed, the normalization of the class (1) proved in Theorem 5 implies that the maximal Lie invariance algebra g f of any equation L f from the class (1) is contained by the algebra g, and the components of any vector field from g are of the form (10). Moreover, for any constant tuple (c 0 , .…”