Symmetry Drivers and Formal Integrals of Hyperbolic Systems
- PDF / 219,963 Bytes
- 10 Pages / 594 x 792 pts Page_size
- 29 Downloads / 162 Views
SYMMETRY DRIVERS AND FORMAL INTEGRALS OF HYPERBOLIC SYSTEMS S. Ya. Startsev
UDC 517.957, 517.956.3, 514.763.8
Abstract. In this paper, we consider symmetry drivers (i.e., operators that map arbitrary functions of one of independent variables into symmetries) and formal integrals (i.e., operators that map symmetries to the kernel of the total derivative). We prove that a hyperbolic system of partial differential equations possesses a complete set of formal integrals if and only if it admits a complete set of symmetry drivers. This assertion is also valid for difference and differential-difference analogs of scalar hyperbolic equations. Keywords and phrases: nonlinear hyperbolic system, Darboux integrability, higher symmetry, conservation law, integral, Laplace invariants. AMS Subject Classification: 37K05, 37K10, 35L70, 39A14, 34K99
1.
Introduction. Let us consider a partial differential equation of the form uxy = F (x, y, u, ux , uy ).
(1)
Mixed partial derivatives of u can be excluded due to the equation. Therefore, without loss of generality, we can assume that all objects related to the equation (such as symmetries and conservation ¯j := ∂ j u/∂y j . The term “function” means a laws) depend only on x, y, u0 := u, ui := ∂ i u/∂xi , and u differential function, i.e., functions considered may depend on a finite number of the above variables. We denote by Dx and Dy the total derivatives with respect to x and y due to Eq. (1). One of the most known integrable equations of the form (1) is the Liouville equation uxy = eu . In the context of this paper, we indicate the following properties of this equation: ¯ ∈ ker Dx that that are not (1) It admits nontrivial x and y-integrals, i.e., functions w ∈ ker Dy and w functions only of x and y (functions depending only on x and y are usually called trivial integrals). ¯ = uyy − u2y /2. In the case of the Liouville equation, w = uxx − u2x /2 and w u (2) The linearization Dx Dy (f ) = e f of the Liouville equation is integrable by the so-called Laplace cascade method of integration (a brief description of this method is given below; for more details see, e.g., [6, 9]). We note that the expression “integrability of the linearization of Eq. (1) by the Laplace method” means not only the termination of the chain of Laplace invariants in both directions, but also the existence of nonzero functions θ ∈ ker(Dy − Fux ) and θ¯ ∈ ker(Dx − Fuy ) (note that for the linearizations of nonlinear equations, the existence of such functions is not guaranteed, in contrast to linear equations). (3) The use of the Laplace cascade method yields the differential operators S and S¯ that map any elements of the kernels Dy and Dx , respectively, to solutions of the linearized equation (i.e., in symmetries, since the linearization of Eq. (1) is the defining relation for its symmetries). Such operators are called symmetry drivers. As was shown in [10], in the case of the Liouville equation S = Dx + ux and S¯ = Dy + uy . The simultaneous presence of these three properties is not accidental: in a number of work
Data Loading...
