The integrable hierarchy and the nonlinear Riemann-Hilbert problem associated with one typical Einstein-Weyl physico-geometric dispersionless system

TL;DR

This paper develops an integrable hierarchy linked to an Einstein-Weyl dispersionless system, using Lax pairs, twistor theory, and a nonlinear Riemann-Hilbert problem to deepen understanding of its solutions.

Contribution

It introduces a new integrable hierarchy with a tau function and twistor structure, and constructs a nonlinear Riemann-Hilbert problem for the system.

Findings

Established an integrable hierarchy from a dispersionless PDE.

Confirmed the existence of a tau function for the hierarchy.

Constructed a nonlinear Riemann-Hilbert problem to analyze solutions.

Abstract

From a specific series of exchange conditions for a one-parameter Hamiltonian vector field, we establish an integrable hierarchy using Lax pairs derived from the dispersionless partial differential equation. An exterior differential form of the integrable hierarchy is introduced, further confirming the existence of the tau function. Subsequently, we present the twistor structure of the hierarchy. By constructing the nonlinear Riemann Hilbert problem for the equation, the structure of the solution to the equation is better understood.