Dispersionless scalar integrable hierarchies, Whitham hierarchy, and the quasiclassical ∂̄-dressing method

Abstract

A note on reductions of the dispersionless Toda hierarchy J. Math. Phys. 51, 122704 (2010) The quasiclassical limit of the scalar nonlocal ‫ץ‬ -problem is derived and a quasiclassical version of the ‫ץ‬ -dressing method is presented. Dispersionless KadomtsevPetviashvili ͑KP͒, modified KP, and dispersionless two-dimensional Toda lattice ͑2DTL͒ hierarchies are discussed as illustrative examples. It is shown that the universal Whitham hierarchy is nothing but the ring of symmetries for the quasiclassical ‫ץ‬ -problem. The reduction problem is discussed and, in particular, the d2DTL equation of B type is derived.

On Dispersionless BKP Hierarchy and its Reductions

“…The quasiclassical∂-dressing scheme for dispersionless KP hierarchy [18,19,20,21] is based on nonlinear Beltrami equation…”

“…In the dispersionless case the study of hierarchies of B type is just in the very beginning [28,20].…”

“…Quasi-classical∂-dressing method, proposed in [18,19,20,21], gives a new approach to study the properties of dispersionless integrable hierarchies, including various addition formulae, symmetry constraints etc. [22,23,24].…”

Looped Cotangent Virasoro Algebra and Non-Linear Integrable Systems in Dimension 2 + 1

“…Another example is the more recent version of the KP equation, namely so-called KP of type B (BKP) (see [18,5]). The dispersionless BKP is of the form u t = α u x u 2 + β u y u + u x ∂ −1…”

On kernel formulas and dispersionless Hirota equations of the extended dispersionless BKP hierarchy

“…In particular we would like to derive dHirota equations of a universal integrable hierarchy underlying topological Landau-Ginzburg models of D-type proposed by Takasaki [23]. Since this integrable hierarchy is an extension of the dispersionless BKP (dBKP) hierarchy [7,22,15,2] and has two sets of time variables we refer it to the extended dispersionless BKP (EdBKP) hierarchy [6]. The EdBKP hierarchy resembles the dToda hierarchy in many formulations such as dressing operators, Orlov functions, Riemann-Hilbert problem, additional symmetries, w-algebras, hodograph solutions etc.…”