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**Title**

**Speaker**

**Time**

**Place**

Refining the invariant subspace method of evolution equations

Wen-Xiu Ma

10:00am-11:00am

LIF 269

**Abstract**

The invariant subspace method is refined, with a view to shedding light on unity and diversity of exact solutions to evolution equations. The crucial idea is to take solution subspaces of linear ordinary differential equations as invariant subspaces that evolution equations admit. A few of examples will be given to illustrate the refined approach.

**Title**

**Speaker**

**Time**

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Mittag-Leffler Functions and Fractional Differential Equations

Magdy Asaad

10:00am-11:00am

LIF 269

**Abstract**

The Mittag-Leffler method has been used in different areas of mathematics. The main aim of the method is to prove solvability of linear fractional differential equations. Moreover, Mittag-Leffler functions with two parameters play a pivotal role and appear regularly in solutions of fractional differential equations. In this talk, we would like to discuss applications of Mittag-Leffler functions in the study of fractional differential equations.

**Title**

**Speaker**

**Time**

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Existence of solutions to the fractional-order diffusion and wave equations

Magdy Asaad

10:00am-11:00am

LIF 269

**Abstract**

Fractional calculus is a collection of relatively little-known mathematical results concerning generalizations of differentiation and integration to fractional orders. While these results have been accumulated over centuries in various branches of mathematics, they have until recently found little appreciation or application in physics and other mathematically oriented sciences. This situation is beginning to change, and there are now a growing number of research areas in physics which employ fractional calculus. We would like to introduce some basics of fractional calculus and discuss the existence of solutions to the fractional-order diffusion and wave equations.

**Title**

**Speaker**

**Time**

**Place**

Vector fields and \(1\)-forms on manifolds

Junyi Tu

10:00am-11:00am

LIF 269

**Abstract**

We would like to discuss some basic structures on real smooth and complex holomorphic manifolds, including vector fields and \(1\)-forms, flows and the straightening theorem.

**Title**

**Speaker**

**Time**

**Place**

Loop algebras and integrable couplings

Wen-Xiu Ma

10:00am-11:00am

LIF 269

**Abstract**

We will talk about integrable couplings of soliton equations. The desired integrability is exhibited through the variational identities over semi-direct sums of Lie algebras. A few illustrative examples are given to show how to generate integrable couplings from loop algebras.

**Title**

**Speaker**

**Time**

**Place**

Hamiltonian structures and bi-Hamiltonian structures of integrable equations

Wen-Xiu Ma

10:00am-11:00am

LIF 269

**Abstract**

We explore Hamiltonian structures and bi-Hamiltonian structures of integrable equations by the zero curvature formulation. The key tool is the trace identity associated with loop algebras. We shed light on some basic aspects of the theory by an often-quoted example — the AKNS soliton hierarchy.

**Title**

**Speaker**

**Time**

**Place**

Hirota bilinear equations and vertex operators

Junyi Tu

10:00am-11:00am

LIF 269

**Abstract**

We will discuss a connection between Hirota bilinear equations and vertex operators. In particular, we will show how it works from the \(1\)-soliton solution to the \(N\)-soliton solution of the KdV equation.

**Title**

**Speaker**

**Time**

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A study on the complex-valued partial differential equations

Netra Khanal

University of Tampa

10:00am-11:00am

LIF 269

**Abstract**

We will discuss the blow-up solutions of complex-valued Burgers and KdV equations and the regularity of series-type solutions of complex KdV-Burgers equation under some mild conditions.

**Title**

**Speaker**

**Time**

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The pseudo-differential operator representation for the hierarchy of KdV equations, Part II

Mengshu Zhang

10:00am-11:00am

LIF 269

**Title**

**Speaker**

**Time**

**Place**

The pseudo-differential operator representation for the hierarchy of KdV equations

Mengshu Zhang

10:00am-11:00am

LIF 269

**Abstract**

We will talk about the higher order KdV equations by considering their pseudodifferential operator representation, and compute infinitely many commuting symmetries through the Lax pairs of pseudodifferential operators.

**Title**

**Speaker**

**Time**

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The KdV equation and its symmetries, Part II

Jinghan Meng

10:00am-11:00am

LIF 269

**Abstract**

We will continue to talk about symmetries of the KdV equation in terms of infinitesimal transformations by a nonlinear evolution equation. The KdV equation is one of typical integrable evolution equations. We will also show how to drive it through the compatibility conditions of two spectral problems.

**Title**

**Speaker**

**Time**

**Place**

The KdV equation and its symmetries

Jinghan Meng

10:00am-11:00am

LIF 269

**Abstract**

We will talk about symmetries of the KdV equation in terms of infinitesimal transformations by a nonlinear evolution equation. The KdV equation is one of typical integrable evolution equations. We will also show how to drive it through the compatibility conditions of two spectral problems.

**Title**

**Speaker**

**Time**

**Place**

Symbolic computation of analytic solutions for nonlinear differential equations

Yinping Liu

East China Normal University

P.R. China

10:00am-11:00am

LIF 269

**Abstract**

In this talk, I will give a brief introduction about symbolic computations on nonlinear differential equations. The outline of my talk is as follows:

- The Elliptic Equation method to construct different types of exact solutions for nonlinear evolution equations, and an automated derivation program.
- An algorithm to construct auto-BTs for given nonlinear evolution equations, and a program for automated derivation of auto-BTs as well as the corresponding superposition formulas.
- A program for automated derivation of analytic approximate solutions for nonlinear differential equations with initial or boundary conditions.
- A frame of a database for differential equations.