The large t behaviour of solutions for a new generalized r-th dispersionless Harry Dym equation

Abstract

Manakov and Santini, etc., have solved inverse scattering problem for dispersionless integrable partial differential equations (PDEs), and used these to construct the formal solutions for integrable equations. In this paper, we derive a new equation from a pair of two-dimensional vector fields, termed the generalized r-th dispersionless Harry Dym (g-rdDym) equation, which reduces to the standard (2+1)-dimensional r-th dispersionless Harry Dym (rdDym) equation. Then we construct large $t$ behaviour of formal solution of Cauchy problem by applying associated Riemann-Hilbert (RH) inverse problem, and describe a new class of particular solutions via the exponential functions. This paper investigates not only the rdDym equation, but also its corresponding generalized equation, i.e. the g-rdDym equation.