Exact Solutions Of Dunajski Hierarchy And The Longtime Behavior Of Its Linear System

As an important method of mathematical physics,inverse scattering transform is mainly applied to study nonlinear partial differential equations.In 2005,Manakov and Santini proposed a new inverse scattering transform method to solve nonlinear integrable partial differential equations of hydrodynamics type by studying on direct problems and inverse problems which are related with Lax pairs arising from one-parameter families of vector fields.With the basis of this new method,this paper discusses exact solutions of Dunajski equation in(3+1)dimension.The equation,along with anti-self-duality field equation proposed by Einstein is of great significance to mathematics and physics.This paper constructs solvable nonlinear Riemann-Hilbert problems concerning to hyperbolic function,discusses nonlinear Riemann-Hilbert problems to study solutions of Dunajski hierarchy through dynamical systems,as well as takes into account long-time behavior of linearized Dunajski equation's solutions.