Dispersionful analog of the Whitham hierarchy

Abstract

The dispersionful analogue, by means of Lax formalism, of the zero-genus universal Whitham hierarchy together with its algebraic orbit finite-field reductions is considered. The theory is illustrated by several significant examples.

Dispersive deformations of hydrodynamic reductions of (2 + 1)D dispersionless integrable systems

“…Several ways to solve the problem of quantization of dispersionless Lax pairs of the form (42) were proposed in the literature, in particular, including the Moyal deformation of the Poisson bracket { , }, see e.g. [33,28,29,3,4]. These methods work well for a limited number of examples, allowing one to reconstruct the admissible dispersive terms, however, they meet difficulties when applied to Lax pairs with a more complicated dependence on ξ.…”

An extension of the Kadomtsev–Petviashvili hierarchy and its hamiltonian structures

“…In this way, when N = 1 one gets a hierarchy of the form (2.12)-(2.13) (up to a nonessential transformation L 1 → L * 1 ). However, the authors of [24] noted that the operators Ω i,n in their construction do not have a "limited" or "compact" form. What is more, Hamiltonian structures for such counterparts seem not have been studied in the literature.…”

“…Let us proceed to check that the equations (2.12)-(2.13) are well defined (cf. [24]). First, it can be seen that the left and the right hand sides of equations in line (2.12) are operators with only negative part, which implies that such equations are well defined.…”

“…The dispersionless extended KP hierarchy (or the Whitham hierarchy of genus zero with one marked point [24,25], without the logarithm flow) is defined by where F, H are arbitrary functionals depending on a ∈ H ϕ , with variational gradients δF/δa, δH/δa respectively (the definition of variational gradient is almost the same as before, so it is omitted here).…”

Hamiltonian Systems of Hydrodynamic Type in 2 + 1 Dimensions

“…This system also arises in the genus zero case of the universal Whitham hierarchy [17,18]; its dispersionful analogue was constructed in [26].…”