The $\bar{\partial}$-approach to the dispersionless KP hierarchy

Abstract

The dispersionless limit of the scalar nonlocal ∂-problem is derived. It is given by a special class of nonlinear first-order equations. A quasi-classical version of the ∂-dressing method is presented. It is shown that the algebraic formulation of dispersionless hierarchies can be expressed in terms of properties of Beltrami tupe equations. The universal Whitham hierarchy and, in particular, the dispersionless KP hierarchy turn out to be rings of symmetries for the quasi-classical ∂-problem.

The characterization of two-component (2+1)-dimensional integrable systems of hydrodynamic type

“…This establishes the equivalence of the two approaches to integrability of (2+1)-dimensional hydrodynamic type systems. The quasi-classical ∂-dressing approach to the solution of (2+1)-dimensional dispersionless systems based on the pseudopotentials of the above type was proposed in the series of recent publications [16,17,18,19,1]. It is not completely clear at the moment how exact solutions describing nonlinear interactions of planar simple waves fit into this scheme.…”

Untitled

“…Several methods for constructing solutions of the dKP hierarchy through S-functions have been devised [3,4,6,8]. We will be here concerned with the∂-method proposed in [13]- [14], which aims to characterize S-functions by means of solutions of∂-equations of the form (1). A main feature of this approach is that the symmetries of (1) (first order variations f := δS) are determined by the family of Beltrami equations (2)- (3).…”

“…In this sense, it is worth-mentioning their connections [11,12] with some classical problems of the theory of conformal maps. Furthermore, we have recently proposed a∂-method [13]- [14] for studying dispersionless integrable hierarchies which reveals an intimate relation between these hierarchies and the theory of quasiconformal mappings [15]- [18].…”

The constrained dispersionless mKP hierarchy and the dispersionless mKP hierarchy with self-consistent sources

“…Recently much attention has been focused on the study of dispersionless integrable hierarchies which have important applications from complex analysis to topological field theory (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]). In [2,3,6,7,9], a standard procedure of dispersionless limit of integrable dispersionfull hierarchies is proposed.…”