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A class of nonlinear problems on the plane, described by nonlinear inhomogeneous ¯ ∂-equations, is considered. It is shown that the corresponding dynamics, generated by deformations of inhomogeneous terms (sources) is described by Hamilton-Jacobi type equations associated with hierarchies of dispersionless integrable systems. These hierarchies are… (More)
The quasi-classical ¯ ∂-dressing method is used to derive compact generating equations for dispersionless hierarchies. Dispersionless Ka-domtsev-Petviashvili (KP) and two-dimensional Toda lattice (2DTL) hierarchies are considered as illustrative examples.
The factorization problem of the multi-component 2D Toda hierarchy is used to analyze the dispersionless limit of this hierarchy. A dispersive version of the Whitham hierarchy defined in terms of scalar Lax and Orlov–Schulman operators is introduced and the corresponding additional symmetries and string equations are discussed. Then, it is shown how KP and… (More)
We present a theory of compatible differential constraints of a hydrodynamic hierarchy of infinite-dimensional systems. It provides a convenient point of view for studying and formulating integrability properties and it reveals some hidden structures of the theory of integrable systems. Illustrative examples and new integrable models are exhibited.
In this paper we discuss the dispersionless limit of the multicomponent 2D Toda hierarchy. The discrete flows of the hierarchy are used to define charge preserving Lax and Orlov–Schulman operators. This construction allows us to perform two types of dispersionless limits, one type leads to the 0-genus universal Whitham hierarchy while the other leads to a… (More)