# Research

## Differential Equations

### Tuesday, December 21, 2010

Title: Conservation laws of equation family with same Kac-Moody-Virasoro symmetry
Speaker: Senyue Lou, Ningbo University
P.R. China
Time: 10:00am‐11:00am
Place: PHY 120

#### Abstract

In this report, we construct conservation laws of the equation family which possesses the same infinite dimensional Kac-Moody-Virasoro algebra as the KP equation. The conservation laws are calculated up to second and third order group invariants and described by many arbitrary functions with various independent arguments.

### Monday, December 13, 2010

Title: Two kinds of infinite dimensional symmetry algebras of the constrained CKP and BKP hierarchies
Speaker: Jingsong He, Ningbo University
P.R. China
Time: 3:00pm‐4:00pm
Place: NES 104

#### Abstract

This talk aims to construct additional symmetries associated with the constrained CKP and BKP hierarchies, and further to give the action of this symmetry flows on the eigenfunction and adjoint eigenfunction. We also show that their acting on the space of the wave operator forms two kinds of centerless Lie algebras — subalgebras of centerless $$W$$-algebras.

### Friday, December 3, 2010

Title: Bilinearization of nonlinear integrable equations, Part IV
Speaker: Yaning Tang, Northwest Polytechnic University
Xi'an, P.R. China
Time: 10:30am‐11:30am
Place: PHY 209 (Lounge)

### Friday, November 19, 2010

Title: Bilinearization of nonlinear integrable equations, Part III
Speaker: Xianqi Li
Time: 10:30am‐11:30am
Place: PHY 209 (Lounge)

### Friday, November 12, 2010

Title: Bilinearization of nonlinear integrable equations, Part II
Speaker: Xianqi Li
Time: 10:30am‐11:30am
Place: PHY 209 (Lounge)

### Friday, November 5, 2010

Title: Bilinearization of nonlinear integrable equations, Part I
Speaker: Xianqi Li
Time: 10:30am‐11:30am
Place: PHY 209 (Lounge)

#### Abstract

We describe procedures for transforming nonlinear partial differential equations, particularly integrable equations, into bilinear forms. A few types of dependent variable transformations will be analyzed and examples include rational, logarithmic and bi-logarithmic transformations. Applications will be made for the KdV equation, the modified KdV equation, the Boussinesq equation and the Kadomtsev-Petviashvili equation and the corresponding soliton solutions will be presented.

### Friday, October 22, 2010

Title: A few Lie algebras and their applications to integrable couplings
Speaker: Yufeng Zhang, Liaoning Normal University
Dalian, P.R. China
Time: 10:30am‐11:30am
Place: PHY 209 (Lounge)

#### Abstract

We introduce four Lie algebras and their corresponding loop algebras. Associated with those loop algebras, three integrable couplings are derived from zero curvature equations by employing the Tu scheme, and their Hamiltonian structures are also obtained by using the variational identity.

### Friday, October 15, 2010

Title: Explicit Flow Equations and Recursion Operator of the ncKP hierarchy, Part II
Speaker: Junyi Tu
Time: 10:30am‐11:30am
Place: PHY 209 (Lounge)

### Friday, October 8, 2010

Title: Explicit Flow Equations and Recursion Operator of the ncKP hierarchy
Speaker: Junyi Tu
Time: 10:30am‐11:30am
Place: PHY 209 (Lounge)

#### Abstract

The explicit expression of flow equations of the noncommutative Kadomtsev-Petviashvili (ncKP) hierarchy is derived. By comparing with the flow equations of the KP hierarchy, our result shows that the additional terms in the flow equations of the ncKP hierarchy consist of commutators of dynamical coordinates $$ui$$, indeed. The recursion operator for the flow equations under the $$n$$-reduction is presented. Further, under $$2$$-reduction, we calculate a nonlocal recursion operator of the noncommutative Korteweg-de Vries hierarchy, which generates a hierarchy of local, higher order flows. Thus we solve the open problem suggested by P. J. Olver and V. V. Sokolov (Commun. Math. Phys. 193 (1998), no. 2, 245-268).

### Friday, September 17, 2010

Title: Index Integral Representations for Connection between Cartesian, Cylindrical, and Spheroidal Systems
Speaker: Sherwin Kouchekian
Time: 10:30am‐11:30am
Place: PHY 209 (Lounge)

#### Abstract

In this talk, we present two index integral representations for connection between Cartesian, cylindrical, and spheroidal coordinate systems in terms of Bessel, MacDonald, and conical functions. Our result is mainly motivated by solution of the boundary value problems in domains composed of both Cartesian and hyperboloidal boundaries, and the need for new integral representations that facilitate the transformation between these coordinates. As a byproduct, the special cases of our results will produce new proofs to known index integrals and provide some new integral identities.

### Friday, September 10, 2010

Title: Symbolic computation of travelling waves of nonlinear PDEs via integrable ODEs
Speaker: Wen-Xiu Ma
Time: 10:30am‐11:30am
Place: PHY 209 (Lounge)

#### Abstract

An algorithm is implemented in Maple for computing travelling wave solutions to nonlinear partial differential equations (PDEs) via integrable ordinary differential equations (ODEs). Taking special integrable ODEs, the algorithm presents various automated exact solution methods such as the extended $$\tanh$$-function method, the Jacobi elliptic-function method and the Bernoulli equation method. A few examples of nonlinear PDEs will be tested to see the symbolic-analytic power of the Maple program.

### Friday, September 3, 2010

Title: Discussion on generating nonlinear integrable couplings
Speaker: Yufeng Zhang, Liaoning Normal University
Dalian, P.R. China
Time: 10:30am‐11:30am
Place: PHY 209 (Lounge)

#### Abstract

Three kinds of Lie algebras are introduced for which the nonlinear integrable couplings of the AKNS hierarchy, the BK hierarchy and the KN hierarchy are obtained, respectively, under the frame of zero curvature equations. The Hamiltonian structures of the nonlinear integrable couplings of the AKNS and KN hierarchies are generated by using the variational identity.