Research On Two Types Of Painlevé Equation Based On Riemann-Hilbert Method

The six types of Painleve equations which now closely related to many mathematical and physical problems.Many analytical,algebraic,and geometric properties are constantly being discovered.The solution of the Painleve equation is called the Painleve function or the Painleve transcendental function,which is a nonlinear special function.The matrix-valued Riemann-Hilbert(RH)numerical framework which is essentially a collocation method.Its core idea is that any curve can be conformally mapped to the unit interval,and it can effectively calculate its value in the unit interval Cauchy transformation.This paper mainly uses the numerical method to solve the approximate solutions of the third and fourth types of Painleve equations,and further verifies the effectiveness and feasibility of the numerical method.First,the method of finding isomonodromy is used to derive the details of the two types of Painleve equations.For the process of the RH problem,the jump matrix and the jump curve are calculated.Secondly,the matrix value RH numerical collocation method is used to conformally map each jump curve of the two types of equations to the unnit interval and express it by Chebyshev basis.The Cauchy transformation of the Chebyshev basis of each jump curve in the unit interval is calculated,and an approximate solution is derived.Finally,Mathematica software is used to draw the solved graph and error graph,and the convergence is analyzed to further verify the feasibility and effectiveness of the method.