Fractional differential generalization of the Burgers hierarchy

S. Yu. Lukashchuk

Ufa University of Science and Technology, Ufa, Russia

Abstract: We generalize the Burgers hierarchy to the case of an arbitrary positive fractional order. We introduce a nonlinear fractional differential operator generated by a fractional power of the recursion operator of the original hierarchy. We show that, as in the integer case, the fractional differential equations of the generalized hierarchy are linearized by the Cole–Hopf transformation. In particular, the fractional differential generalization of the Burgers equation is transferred by this transform into a fractional differential superdiffusion equation. We find recursion operators for these equations and construct higher symmetries, local and nonlocal, including fractional differential ones.

Keywords: Burgers hierarchy, fractional power of an operator, fractional differential equation, Marchaud fractional derivative, recursion operator, symmetry.

Received: 21.07.2025
Revised: 21.07.2025

DOI: 10.4213/tmf11054

 English version: Theoretical and Mathematical Physics, 2025, 225:3, 2077–2088

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