[Paper Review] Infinite-dimensional symmetries of a two-dimensional generalized Burgers equation

F. Güngör|ArXiv.org|Feb 24, 2009

Nonlinear Waves and Solitons4 references3 citations

TL;DR

This paper determines the conditions under which a two-dimensional generalized Burgers equation with nine arbitrary coefficient functions admits an infinite-dimensional symmetry algebra. Using equivalence transformations, the authors derive a canonical form and show that while Kac-Moody-type symmetries are possible, no Virasoro algebra can be realized, indicating that such variable-coefficient generalizations remain non-integrable despite large symmetry groups.

ABSTRACT

The conditions for a generalized Burgers equation which a priori involves nine arbitrary functions of one, or two variables to allow an infinite dimensional symmetry algebra are determined. Though this algebra can involve up to two arbitrary functions of time, it does not allow a Virasoro algebra. This result confirms that variable coefficient generalizations of a non-integrable equation should be expected to remain as such.

Motivation & Objective

• To determine the conditions under which a two-dimensional generalized Burgers equation with nine arbitrary coefficient functions admits an infinite-dimensional symmetry algebra.
• To investigate whether such equations can support a Virasoro algebra, a hallmark of integrability in 2+1 dimensions.
• To classify all cases of infinite-dimensional symmetry realization, particularly focusing on Kac-Moody and Abelian structures.
• To apply symmetry reduction techniques to construct explicit solutions from the infinite-dimensional symmetry algebras.
• To clarify the distinction between integrable equations (like KP) and non-integrable generalizations (like Burgers) via symmetry structure.

Proposed method

• The authors use equivalence (allowed) transformations to reduce the general equation to a canonical form, preserving the structure of coefficient functions.
• They derive determining equations for symmetries by analyzing the invariance of the canonical generalized Burgers equation under point transformations.
• The symmetry algebra is analyzed by solving the determining equations, focusing on cases with arbitrary functions of time or space.
• The paper examines the possibility of time reparametrization invariance and Kac-Moody algebra realization through explicit construction of vector fields.
• Symmetry reduction is applied to invariant solutions, reducing the PDE to ODEs or simpler PDEs using group-invariant ansatzes.
• Specific solution families are constructed via ansatzes based on symmetry generators, including moving-frame and non-Abelian Kac-Moody symmetries.

Experimental results

Research questions

• RQ1 Under what conditions does the two-dimensional generalized Burgers equation admit an infinite-dimensional symmetry algebra?
• RQ2 Can the symmetry algebra of the generalized Burgers equation contain a Virasoro subalgebra, indicating integrability?
• RQ3 What is the maximal possible symmetry algebra structure (e.g., Kac-Moody, Abelian) for such equations?
• RQ4 How do the coefficient functions constrain the existence of non-Abelian or Abelian infinite-dimensional symmetries?
• RQ5 Can symmetry reduction yield explicit solutions for the generalized Burgers equation?

Key findings

• The canonical generalized Burgers equation admits an infinite-dimensional Abelian symmetry algebra if and only if the function f(y,t) is identically zero and a, b, c are arbitrary.
• A non-Abelian Kac-Moody symmetry algebra exists when specific coefficient conditions are met, including b(y,t) = b₁(t)y + b₀(t) and c(y,t) = c₂(t)y² + c₁(t)y + c₀(t), with a = f = 0.
• No Virasoro algebra can be realized as a subalgebra of the symmetry algebra, even in the most symmetric cases, confirming that the equation remains non-integrable despite infinite-dimensional symmetry.
• The most general symmetry is the transformation to an arbitrary frame moving in the x-direction, which requires f(y,t) ≡ 0 and leads to solutions linear in x with specific y-dependence.
• Symmetry reduction leads to a reduced PDE of the form (F_t + FF_z + F_zz)_z + ε(ξ²/η²)F_zz + ... = 0, which reduces to the 1D Burgers equation when c(t) = 0.
• Solutions of the 1D Burgers equation can be lifted to 2D via the ansatz (6.5), providing a systematic method to generate y-dependent solutions for the generalized equation.