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Title: The Riemann-Hilbert method for integrable equations Speaker: Wen-Xiu Ma Time: 2:00pm‐3:00pm Place: NES 104
We will talk about the Riemann-Hilbert method for solving integrable equations. The starting point is a pair of matrix spectral problems, called a Lax pair. Soliton solutions can be generated from Riemann-Hilbert problems on the real axis with an identity jump matrix. An illustrative example, associated with an arbitrary order matrix spatial spectral problem, will be presented.
Title: Lump and interaction solutions to an extended \((2+1)\)-dimensional Boussinesq equation Speaker: Hui Wang, Shanghai Maritime University Shanghai, China Time: 2:00pm‐3:00pm Place: NES 104
I will briefly talk about lump and interaction solutions to nonlinear partial differential equations. The example I will focus on is an extended \((2+1)\)-dimensional Boussinesq equation, whose lump and interaction solutions will be explored.
Title: Derive longtime asymptotics of the MKdV equation by Painlevé II Speaker: Fudong Wang Time: 2:00pm‐3:00pm Place: NES 104
We will discuss longtime asymptotics for the MKdV equation by the isomonodromy deformation of the second Painlevé equation.
Title: On some 2D orthogonal \(q\)-polynomials Speaker: Ruiming Zhang, Northwest A&F University PR China Time: 2:00pm‐3:00pm Place: NES 104
This talk is based on a joint work with Professor Mourad E. H. Ismail recently published on AMS Transactions, in which we have generalized several families of 2D orthogonal polynomials used in probability theory and physics (a.k.a. Zernike polynomials). In this talk, I introduce two \(q\)-analogues of the 2D-Hermite polynomials that are polynomials in two complex variables. For both families, I present explicit formulas, raising and lowering operator relations, generating functions, and Rodrigues formulas. These \(q\)-orthogonal polynomials also have unexpected connections to the theory of integer partitions.
Title: The factorization problem on a fixed cross Speaker: Yuan Zhou Time: 2:00pm‐3:00pm Place: NES 104
The factorization problem on a fixed cross is solved explicitly in terms of parabolic cylinder functions. The solution is then used to present the longtime asymptotics of the mKdV equation.
Title: Evaluating integrals on two crosses in oscillatory Riemann-Hilbert problems by the steepest descent method Speaker: Xiang Gu Time: 2:00pm‐3:00pm Place: NES 104
With the help of the Cauchy operators, we will discuss how to separate contributions from two crosses in a contour, and explore long-time asymptotic behavior of the corresponding integrals.
Spring Break — no seminar this week.
Title: Steepest descent method for oscillatory Riemann-Hilbert problems with truncated contours II Speaker: Fudong Wang Time: 2:00pm‐3:00pm Place: NES 104
Title: Steepest descent method for oscillatory Riemann-Hilbert problems with truncated contours Speaker: Fudong Wang Time: 2:00pm‐3:00pm Place: NES 104
We will discuss how to use the Cauchy operator and apply its properties, say, \(L^2\) boundedness, to convert argumented Riemann-Hilbert problems to truncated Riemann-Hilbert problems with controlled error terms.
Title: Studies on exact solutions of partial differential and differential-difference equations Speaker: Yongli Sun, Beijing University of Chemical Technology PR China Time: 2:00pm‐3:00pm Place: NES 104
In this seminar, I will briefly introduce my recent research. One is about exact solutions of some partial differential equations, based on the transformed rational function method. The other is about soliton solutions to the Blaszak-Marciniak lattice II, by the Darboux transformation.
Title: A matrix spectral problem based on \(\mathrm{so}(3,R)\) and its associated soliton hierarchy Speaker: Sumayah Batwa Time: 2:00pm‐3:00pm Place: NES 104
In this talk, I will discuss a spectral problem associated with the real Lie algebra \(\mathrm{so}(3,R)\) and generate a hierarchy of soliton equations from zero curvature equations linked with the spectral problem. Moreover, I will give an illustrative example of soliton equations with a bi-Hamiltonian structure.
Title: Contour deformation in oscillatory Riemann-Hilbert problems associated with the mKdV equation III Speaker: Fudong Wang Time: 2:00pm‐3:00pm Place: NES 104
Title: Nonlocal symmetries with pseudopotentials Speaker: Xiazhi Hao, East China Normal University Time: 2:00pm‐3:00pm Place: NES 104
In this talk, I will discuss nonlocal symmetries by including pseudopotentials, which are equivalent to Lie point symmetries of a prolonged system. Moreover, I will show that through the transformations resulted from nonlocal symmetries, exact solutions to the original system can be generated.
Title: Contour deformation in oscillatory Riemann-Hilbert problems associated with the mKdV equation II Speaker: Fudong Wang Time: 2:00pm‐3:00pm Place: NES 104
Title: Contour deformation in oscillatory Riemann-Hilbert problems associated with the mKdV equation Speaker: Fudong Wang Time: 2:00pm‐3:00pm Place: NES 104
I will discuss about how to extend oscillatory Riemann-Hilbert problems to an augmented contour. The focus will be on how to apply the steepest descent method introduced by Percy Deift and Xin Zhou to analyze long time asymptotics for the mKdV equation.