A classification is given of equations vtt=f(v,vx)vxx+g(x,vx) admitting an extension by one of the principal Lie algebra of the equation under consideration. The paper is one of few applications of a new algebraic approach to the problem of group classification: the method of preliminary group classification. The result of the work is a wide class of equations summarized in Table II.
In this paper we generalize the classification of self-adjoint second order linear partial differential equation to a family of nonlinear wave equations with two independent variables. We find a class of quasi self-adjoint nonlinear equations which includes the self-adjoint linear equations as a particular case. The property of a differential equation to be quasi self-adjoint is important, e.g. for constructing conservation laws associated with symmetries of the differential equation.
In this paper we consider a (2 + 1) dimensional generalized Burgers equation. After having written this equation as a system in two dependent variables, we show that it is quasi self-adjoint and find a nontrivial additional conservation law for this system.
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