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Date & Time: 3 September 2009, 11.00am

Organization: Institute of Mathematics, Statistics & Actuarial Science, University of Kent, Canterbury, CT2 7NF, UK

Venue: Link Building Seminar Room Building 60, RSPhysSE, ANU

Title: Rational Solutions of Soliton Equations and Applications to Vortex Dynamics

Abstract: In this talk, I shall discuss special polynomials associated with rational solutions for the Painleve equations and of the soliton equations which are solvable by the inverse scattering method, including the Korteweg-de Vries, Boussinesq and nonlinear Schrodinger equations. The Painleve equations are six nonlinear ordinary dierential equations that have been the subject of much interest in the past thirty years, which have arisen in a variety of physical applications. Further they may be thought of as nonlinear special functions. Rational solutions of the Painleve equations are expressible in terms of the logarithmic derivative of certain special polynomials. For the second Painleve equation (PII) these polynomials are known as the Yablonskii-Vorob'ev polynomials, rst derived in the 1960's by Yablonskii and Vorob'ev. The locations of the roots of these polynomials is shown to have a highly regular triangular structure in the complex plane. The analogous special polynomials associated with rational solutions of the fourth Painleve equation (PIV), which are known as the generalized Hermite polynomials and generalized Okamoto polynomials, are described and it is shown that their roots also have a highly regular structure. The Yablonskii-Vorob'ev polynomials arise in string theory and the generalized Hermite polynomials in the theories of random matrices and orthogonal polynomials. It is well known that soliton equations have symmetry reductions which reduce them to the Painleve equations, e.g. scaling reductions of the Korteweg-de Vries equation is expressible in terms of PII and scaling reductions of the Boussinesq and nonlinear Schrodinger equations are expressible in terms of PIV. Hence rational solutions of these soliton equations can be expressed in terms of the Yablonskii and Vorob'ev, generalized Hermite and generalized Okamoto polynomials. Further general rational solutions of equations for the Korteweg-de Vries, Boussinesq equations and nonlinear Schrodinger equations, which involve arbitrary parameters, will also be described. I shall also discuss applications of these special polynomials associated with rational solutions for the Painleve and soliton equations to point vortex dynamics. Further multivortex solutions of the complex Sine-Gordon equation on the plane will be expressed in terms of special polynomials associated with rational solutions of the fifth Painleve equation, which are expressed as double wronskians of associated Laguerre polynomials.

 

 

Date & Time: 3 July 2008, 2.00 pm

Organization: Centre National de la Recherche Scientique Laboratoire de Physique de la Matiere Condensee Universite de Nice - Sophia Antipolis, Parc Valrose, F-06108 Nice Cedex 2, France

Venue: Link Building Seminar Room Building 60, RSPhysSE, ANU

Title: Backward Three-Wave Interactions in Optical Parametric Oscillators

Abstract: Recent experimental demonstration of a backward mirrorless optical parametric oscillator opens the way for achieving two non-degenerate three-wave interaction processes in Optical Parametric Oscillators: (1) ultra-coherent output from an incoherent pump pulse where the incoherence of the pump is absorbed by the co-propagating wave moving at the same group-velocity of the pump, or (2) three-wave ps soliton morphogenesis where a cw-pump generates a symbiotic backward dissipative soliton when interacting with both counterpropagating signal and idler waves.

 

 

Date & Time: 10 December 2004, 11.00am

Organization: Australian Nuclear Science and Technology Organisation, PMB 1, Menai (Sydney), NSW, 2234, Australia

Venue: Link Building Seminar Room Building 60, RSPhysSE, ANU

Title: Nonlinear waves in systems with low- and high-frequency dispersions

Abstract: A new model equation describing wave propagation in the verity of physical systems was derived for the first time by L.A. Ostrovsky in 1978: (u_t + _c0_ux + αuu_x + βu_xxx)x = γu. The equation generalises the well-known KdV equation and contains both high-frequency dispersion (the term ~ β) and low-frequency dispersion (the term ~ γ). The reduced version of this equation with β = 0 is valuable itself and represents, to certain extent, the counterpart of KdV equation combining the nonlinearity of a hydrodynamic type and low-frequency dispersion. The full class of its stationary solutions are presented and analysed in detail by means of phase plane and mechanical analogy. It is shown that there are periodic and solitary solutions. Among them there are compound solitons (peakons, smooth-crest and loop solitons), as well as compactons (solitons defined on a compact supports). Structural stability of solitary wave solutions is studied and it is demonstrated that some of them are stable while others destroy under small perturbation of system parameters. For the full Ostrovsky equation the existence of solitary wave solutions depends on the sign of the product βγ. It is shown that stationary solitary wave solution are impossible for βγ > 0, while for βγ < 0 there is a family of solitons with zero total "mass" and oscillating tails. They can form compound solitons and stochastic chains of solitons. Some examples of nonstationary solutions of the full Ostrovsky equation are also presented and discussed.

 

 

Date: 26 November 2004

Organization: Service de physique de l'état condense, CEA Saclay, France

Title: Meromorphic general solution of the seven Hénon-Heiles Hamiltonians

Abstract: In physical autonomous Hamiltonians (kinetic plus potential energy), the requirement that the general solution be a singlevalued function of time strongly constrains the potential. For one degree of freedom, the result is classical: the potential must be an arbitrary polynomial of degree at most four, and the general solution is then elliptic, i.e. meromorphic doubly periodic. For two degrees of freedom, the potential (denoted HÈnon-Heiles from celestial mechanics) must be a precise polynomial in (q₁,q₂) of degree at most four, with possibly some additional inverse square terms, and only seven cases are allowed (selected by the PainlevÈ test) which may have a singlevalued general solution. Some cases have already been integrated. We show here that all the seven cases have a meromorphic general solution explicitly given by a genus two hyperelliptic function.