Lecture By Prof. Xianguo Geng of Zhengzhou University and Prof. Bo Xue of Zhengzhou University​
time: 2021-05-07

Title1: Tetragonal curves and algebro-geometric solutions to the Satsuma-Hirota coupled hierarchy

Speaker:   Prof. Xianguo Geng  ( Zhengzhou University )

Time: Thur, May.6 2021, PM: 15:00-16:00

Location: Tencent Conference

Meeting  Number: 350 271 734

Title2: Some Applications of the steepest descent method to long-time asymptotics

Speaker:   Prof. Bo Xue  ( Zhengzhou University )

Time: Thur, May.6 2021, PM: 16:00-17:00

Location: Tencent Conference

Meeting  Number: 350 271 734

Inviter: Prof. Liming Lin


     On the basis of the characteristic polynomials of Lax matrixes for the soliton hierarchies, we introduce the corresponding algebraic curves, including the hyperelliptic curve, trigonal curve, and tetragonal curve. We study the calculation of genus of algebraic curve, properties at infinity, and the construction of three kinds of Abel differentials. We establish the corresponding Baker-Akhiezer functions and meromorphic functions. The straightening out of various soliton flows is exactly given through the Abel map and Abel-Jacobi coordinates. Using the theory of algebraic curves, we obtain the explicit Riemann theta function representations of the Baker-Akhiezer function and the meromorphic function. As an illustration, we arrive at algebro-geometric solutions of the entire Satsuma-Hirota coupled hierarchy.


     The initial value problem for the Sasa–Satsuma equation is transformed to a 3×3 matrix Riemann–Hilbert problem with the help of the corresponding Lax pair. Two distinct factorizations of the jump matrix and a decomposition of the vector-valued function are given, from which the long-time asymptotics for the Sasa–Satsuma equation with decaying initial data is obtained by using the nonlinear steepest descent method .