Integrable Systems Of Hydrodynamic Type Research Articles

INTEGRABILITY OF THE EQUATIONS FOR NONSINGULAR PAIRS OF COMPATIBLE FLAT METRICS

We solve the problem of describing all nonsingular pairs of compatible flat metrics (or, in other words, nonsingular flat pencils of metrics) in the general N-component case. This problem is equivalent to the problem of describing all compatible Dubrovin–Novikov brackets (compatible nondegenerate local Poisson brackets of hydrodynamic type) playing an important role in the theory of integrable systems of hydrodynamic type and also in modern differential geometry and field theory. We prove that all nonsingular pairs of compatible flat metrics are described by a system of nonlinear differential equations that is a special nonlinear differential reduction of the classical Lame equations, and we present a scheme for integrating this system by the method of the inverse scattering problem. The integration procedure is based on using the Zakharov method for integrating the Lame equations (a version of the inverse scattering method).

Compatible Dubrovin-Novikov Hamiltonian operators, Lie derivative and integrable systems of hydrodynamic type

We prove that two Dubrovin–Novikov Hamiltonian operators are compatible if and only if one of these operators is the Lie derivative of the other operator along a certain vector field. We consider the class of flat manifolds, which correspond to arbitrary pairs of compatible Dubrovin–Novikov Hamiltonian operators. Locally, these manifolds are defined by solutions of a system of nonlinear equations, which is integrable by the method of the inverse scattering problem. We construct the integrable hierarchies generated by arbitrary pairs of compatible Dubrovin–Novikov Hamiltonian operators.

Compatible and Almost Compatible Pseudo-Riemannian Metrics

In the present paper, the notions of compatible and almost compatible Riemannian and pseudo-Riemannian metrics are introduced. These notions are motivated by the theory of compatible Poisson structures of hydrodynamic type (local and nonlocal) and generalize the notion of flat pencils of metrics, which plays an important role in the theory of integrable systems of hydrodynamic type and Dubrovin's theory of Frobenius manifolds. Compatible metrics generate compatible Poisson structures of hydrodynamic type (these structures are local for flat metrics and nonlocal if the metrics are not flat). For the “nonsingular” case in which the eigenvalues of a pair of metrics are distinct, we obtain a complete explicit description of compatible and almost compatible metrics.

Nonhomogeneous hydrodynamic-type equations and their superextension

It is observed that some (x,t) interchanged nonlinear integrable system do have the characteristics of nonhomoneous integrable system of hydrodynamic type. The second Hamiltonian structure obtained through the Miura map or Lie derivative with respect to the master symmetry do have the requisite features. In the present communication we study the cases of dispersive water wave and NLS equation with (x,t) interchanged. Along with these we also analyse the case of supersymmetric version of the (x,t) interchanged KdV problem which yields an example of a super hydrodynamics system.

Separable Hamiltonians and integrable systems of hydrodynamic type

We exhibit a surprising relationship between separable Hamiltonians and integrable, linearly degenerate systems of hydrodynamic type. This gives a new way of obtaining the general solution of the latter. Our formulae also lead to interesting canonical transformations between large classes of Stäckel systems.

Differential geometry of strongly integrable systems of hydrodynamic type

Here the matrix (gij) (assumed nondegenerate) defines a pseudo-Riemannian metric (with upper indices) of zero curvature on the u-space, Fjk i = Fjki(u) being the corresponding Levi-Civita connection. Thus, the integrability condition can be formulated in terms of the differential geometry of SHT. For such integrable systems S. P. Tsarev [3] found a generalization (for N _> 3) of the hodograph method which lets one "linearize" the system and thus in some sense "integrate" it. More precisely, let the system (i) in the coordinates u I ..... u N be already diagonal: u~, ~ = t , . . . , N , (4 )