Breaking solitons. Systems of hydrodynamic type

A certain class of integrable (n + 1)-dimensional equations was studied by F. Calogero and A. Degasperis by using the generalized Wronskian relations. A general Lax type operator equation was proposed by V.E. Zakharov for constructing (n+ 1)-dimensional integrable equations. These constructions were discussed also in the monograph by R.K. Dodd, J.C. Enbeck, J.D. Gibbon and G.C. Morris. The equations which are studied in this paper are not integrable for the general initial data, but their N-soliton solutions may be found explicitly and they possess the breaking behavior. We consider the differential equations, which are equivalent to the following equation in space of linear operators L and A: Lt = P(L) + nΣk=1 Rk(L, Lyk) + [L, A], where P(L) and Rk(L, Lyk) are certain meromorphic functions of operator L, functions of Rk(L, Lyk) are linear with respect to Lyk. The authors assume that operators L and A depend on the variable t, y1,...,yn and Lyk = ∂L/∂yk. L and A are supposed to be n x n matrices or 1-dimensional differential operators (in the last case L is self-adjoint operator, A is skew-symmetric operator)