Reduction operators of Burgers equation

Abstract

The solution of the problem on reduction operators and nonclassical reductions of the Burgers equation is systematically treated and completed. A new proof of the theorem on the special “no-go” case of regular reduction operators is presented, and the representation of the coefficients of operators in terms of solutions of the initial equation is constructed for this case. All possible nonclassical reductions of the Burgers equation to single ordinary differential equations are exhaustively described. Any Lie reduction of the Burgers equation proves to be equivalent via the Hopf–Cole transformation to a parameterized family of Lie reductions of the linear heat equation.

Enhanced Symmetry Analysis of Two-Dimensional Burgers System

“…The classical Burgers equation u t + uu x = νu xx and its 'inviscid' counterpart u t + uu x = 0 admit only trivial common solutions that are at most affine in x. The set of common solutions is exhausted by two solution families [35], the one-parameter family of constant solutions and the two-parameter family of solutions of the form u = (x + C 1 )/(t + C 2 ), where C 1 and C 2 are arbitrary constants. For the 'viscid' Burgers system (1) and the 'inviscid' Burgers system…”

“…Lie reduction 1.8 is G-equivalent to the case w 1 = 0 givingũ x = 0. An alternative way for deriving the system (35) is to make the Hopf-Cole-type transformation u 0 = w 0 /ṽ, u 1 = w 1 /ṽ, v = −2ṽ y /ṽ in the system (32).…”

Extended symmetry analysis of generalized Burgers equations

“…Later, reduction operators of the Burgers equation were objects of study and discussion in a number of papers [4,5,33,36,53]. These studies were summed up in [42]. Attempts to describe nonclassical symmetries for equations from the class (1) with nonconstant f 's were started in [59] for the subclass of equations with f x = 0.…”

“…Here we arrange the consideration of reduction operators for generalized Burgers equations presented in [41,42] and extend it with complete proofs of the corresponding assertions.…”

“…where the component ξ is already set to 1 using the equivalence relation of reduction operators; cf. [42,Section 2] for the Burger equation L 1 .…”