Lie Symmetry Analysis of the Hopf Functional-Differential Equation

Abstract

In this paper, we extend the classical Lie symmetry analysis from partial differential equations to integro-differential equations with functional derivatives. We continue the work of Oberlack and Wacławczyk (2006, Arch. Mech. 58, 597), (2013, J. Math. Phys. 54, 072901), where the extended Lie symmetry analysis is performed in the Fourier space. Here, we introduce a method to perform the extended Lie symmetry analysis in the physical space where we have to deal with the transformation of the integration variable in the appearing integral terms. The method is based on the transformation of the product y(x)dx appearing in the integral terms and applied to the functional formulation of the viscous Burgers equation. The extended Lie symmetry analysis furnishes all known symmetries of the viscous Burgers equation and is able to provide new symmetries associated with the Hopf formulation of the viscous Burgers equation. Hence, it can be employed as an important tool for applications in continuum mechanics.

Comments on Janocha et al. Lie Symmetry Analysis of the Hopf Functional-Differential Equation. Symmetry 2015, 7, 1536–1566

“…[10], the general solution of the deterministic Burgers and, thus, also of the statistical Hopf-Burgers equation can be formally expressed in closed form; attempts to evaluate these solutions can be found, e.g., in [11,12]). The consequence for the final result in [1]: the determined symmetries X phys 4 -X phys 6 listed in Table 2 [1] (p. 1562) are, in actual fact, not symmetries, i.e., these Lie-group transformations are not admitted as symmetries by the considered functional Hopf-Burgers equation (the proof is given in Appendix A). Before we reveal all errors of this approach in [1], it is necessary to first clarify two essential points:…”

“…The consequence for the final result in [1]: the determined symmetries X phys 4 -X phys 6 listed in Table 2 [1] (p. 1562) are, in actual fact, not symmetries, i.e., these Lie-group transformations are not admitted as symmetries by the considered functional Hopf-Burgers equation (the proof is given in Appendix A). Before we reveal all errors of this approach in [1], it is necessary to first clarify two essential points:…”

“…However, as we will show in this section, this third approach, as it is used throughout in [1] to systematically determine all Lie-point symmetries of the functional Hopf-Burgers equation in physical space, is based on a wrong fundamental assumption and several subsequent reasoning errors. (Remark: Note that via the Cole-Hopf transformation, cf.…”

“…It is claimed that this new, third approach allows for a standard Lie-point symmetry analysis without having to directly transform the volume element of integration. Listed as the third item in [1] (p. 1549), this approach is intended to avoid the "more complicated" [1] (p. 1549) approaches of Ibragimov et al (which rests on a Lie-Bäcklund analysis) and that of Fushchich and Zawistowski et al (which incorporates the transformation of the volume element (Jacobian) within a standard Lie-point analysis), when specifically extended to functional equations in the sense as described in [1].…”

“…Next to the two different and well-established approaches of Ibragimov et al [2,3] and of Fushchich and Zawistowski et al [4][5][6] to systematically determine Lie symmetries of integro-differential equations, the latest study of Janocha et al [1] claims to have found a third approach in how to perform a Lie-group symmetry analysis for such equations, in particular as to how this approach can be applied to integro-differential equations of the functional type (a general introduction into the theory of Lie-group symmetry analysis can be found, e.g., in [7][8][9]). It is claimed that this new, third approach allows for a standard Lie-point symmetry analysis without having to directly transform the volume element of integration.…”