Symmetries of linear and linearizable systems of differential equations
Two sorts of integrable partial differential equations in two independent variables \(t,x\) are considered. A Lax type equation is one that may be written as \(L_t = [L,A]\), where \(L,A\) are ordinary differential operators in the variable \(x\). A Burgers type equation is then introduced as one that is of the form \(L_t = [L,A]\) modulo the left principal ideal generated by \(L\) (these equations are shown to be linearizable by a generalized Hopf-Cole transformation). Recurrent formulas for coefficients of the operator \(A\) are found. Infinite-dimensional algebras of symmetries are constructed for both types of equations. Earlier work on symmetries of ordinary differential equations by the same author [Transl., Ser 2, Am. Math. Soc. 167(24), 193-206 (1995; Zbl 0837.58025)] is exploited. The first part of the paper repeats the essential results, notably the explicit isomorphism between the symmetry algebra and the algebra of vector fields on the space of solutions. The subalgebra of point symmetries in the general case and the algebra of contact symmetries in the case of one second-order ODE are described in detail.
scientific article; zbMATH DE number 1340265
scientific article