For the degenerate Burgers equation u_t + u u_x - u_yy = 0, all generalized symmetries reduce to Lie symmetries, and conservation laws are in one-to-one correspondence with solutions of the backward heat equation.
Group classification and exact solutions of variable-coefficient generalized Burgers equations with linear damping
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abstract
Admissible point transformations between Burgers equations with linear damping and time-dependent coefficients are described and used in order to exhaustively classify Lie symmetries of these equations. Optimal systems of one- and two-dimensional subalgebras of the Lie invariance algebras obtained are constructed. The corresponding Lie reductions to ODEs and to algebraic equations are carried out. Exact solutions to particular equations are found. Some generalized Burgers equations are linearized to the heat equation by composing equivalence transformations with the Hopf-Cole transformation.
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Extended symmetry analysis of two-dimensional degenerate Burgers equation
For the degenerate Burgers equation u_t + u u_x - u_yy = 0, all generalized symmetries reduce to Lie symmetries, and conservation laws are in one-to-one correspondence with solutions of the backward heat equation.