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Mathematics & Statistics

(Leader: Prof. Wen-Xiu Ma)

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**Speaker**

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Conservation laws of equation family with same Kac-Moody-Virasoro symmetry

Senyue Lou

Ningbo University

P.R. China

10:00am-11:00am

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.

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**Speaker**

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Two kinds of infinite dimensional symmetry algebras of the constrained CKP and BKP hierarchies

Jingsong He

Ningbo University

P.R. China

3:00pm-4:00pm

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.

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Bilinearization of nonlinear integrable equations, Part IV

Yaning Tang

Northwest Polytechnic University

Xi'an, P.R. China

10:30am-11:30am

PHY 209 (Lounge)

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Bilinearization of nonlinear integrable equations, Part III

Xianqi Li

10:30am-11:30am

PHY 209 (Lounge)

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Bilinearization of nonlinear integrable equations, Part II

Xianqi Li

10:30am-11:30am

PHY 209 (Lounge)

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Bilinearization of nonlinear integrable equations, Part I

Xianqi Li

10:30am-11:30am

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.

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A few Lie algebras and their applications to integrable couplings

Yufeng Zhang

Liaoning Normal University

Dalian, P.R. China

10:30am-11:30am

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.

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Explicit Flow Equations and Recursion Operator of the ncKP hierarchy, Part II

Junyi Tu

10:30am-11:30am

PHY 209 (Lounge)

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Explicit Flow Equations and Recursion Operator of the ncKP hierarchy

Junyi Tu

10:30am-11:30am

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).

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Index Integral Representations for Connection between Cartesian, Cylindrical, and Spheroidal Systems

Sherwin Kouchekian

10:30am-11:30am

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.

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Symbolic computation of travelling waves of nonlinear PDEs via integrable ODEs

Wen-Xiu Ma

10:30am-11:30am

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.

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Discussion on generating nonlinear integrable couplings

Yufeng Zhang

Liaoning Normal University

Dalian, P.R. China

10:30am-11:30am

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.