Nonlinear equations in private derivatives, related to the operator of Dirak

Abstract

Theory of integrable nonlinear equations possessing soliton solutions of a new type - tipper solitons. The operator examines the design proposed by O.I. Bogoyavlensky, and having attractors in the phase space. For output of a new nonlinear equation is used operator structure Li = [L,A] + P(L), that extends the design of lax, L,A - differential operators, P(L) - polynomial 1-th order. As the operator L, one considers the dierential Dirac operator of the first kind. Are defined by necessary and sufficient conditions under which the operator equation is the compatibility condition for the three linear differential equations: the irst is the eigenvalue equation of the operator L on the space variables and the spectral values parametrically dependent on time, the second describes the dynamics of the eigenfunctions of the operator L in a temporary variable, and the third one deines the spectral function. It is shown that the spectral function can have an orbit -stable subvariety or attractor.