Mathematics / Matematik
Permanent URI for this collectionhttps://hdl.handle.net/11147/8
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Article Citation - WoS: 6Citation - Scopus: 6Integrable Hierarchies and Information Measures(IOP Publishing Ltd., 2008) Parwani, Rajesh R.; Pashaev, OktayIn this paper we investigate integrable models from the perspective of information theory, exhibiting various connections. We begin by showing that compressible hydrodynamics for a one-dimensional isentropic fluid, with an appropriately motivated information theoretic extension, is described by a general nonlinear Schrödinger (NLS) equation. Depending on the choice of the enthalpy function, one obtains the cubic NLS or other modified NLS equations that have applications in various fields. Next, by considering the integrable hierarchy associated with the NLS model, we propose higher order information measures which include the Fisher measure as their first member. The lowest members of the hierarchy are shown to be included in the expansion of a regularized Kullback-Leibler measure while, on the other hand, a suitable combination of the NLS hierarchy leads to a Wootters type measure related to a NLS equation with a relativistic dispersion relation. Finally, through our approach, we are led to construct integrable relativistic NLS equations.Article Citation - WoS: 29Citation - Scopus: 41Soliton resonances in a generalized nonlinear Schrödinger equation(IOP Publishing Ltd., 2008) Pashaev, Oktay; Lee, Jyh Hao; Rogers, ColinIt is shown that a generalized nonlinear Schrödinger equation proposed by Malomed and Stenflo admits, for a specific range of parameters, resonant soliton interaction. The equation is transformed to the 'resonant' nonlinear Schrödinger equation, as originally introduced to describe black holes in a Madelung fluid and recently derived in the context of uniaxial wave propagation in a cold collisionless plasma. A Hirota bilinear representation is obtained and soliton solutions are thereby derived. The one-soliton solution interpretation in terms of a black hole in two-dimensional spacetime is given. For the two-soliton solution, resonant interactions of several kinds are found. The addition of a quantum potential term is considered and the reduction is obtained to the resonant NLS equation. © 2008 IOP Publishing Ltd.Article Citation - WoS: 4Citation - Scopus: 4The Cauchy Problem for the Planar Spin-Liquid Model(IOP Publishing Ltd., 2005) Pashaev, Oktay; Chang, Nai-HengIn this paper, we study the Cauchy problem of a two-dimensional model for a moving ferromagnetic continuum and prove global existence and uniqueness of solutions. In addition, equivalence to the coupled system of nonlinear Schrödinger equations interacting with a Chern-Simons gauge field is established.Article Citation - WoS: 10Citation - Scopus: 9Q-Deformed and C-Deformed Harmonic Oscillators(Yukawa Institute for Theoretical Physics, 2003) Sogami, Ikuo S.; Koizumi, Kouzou; Mir-Kasimov, Rufat M.Hamilton functions of classical deformed oscillators (c-deformed oscillators) are derived from Hamiltonians of g-deformed oscillators of the Macfarlane and Dubna types. A new scale parameter, lq, with the dimension of length, is introduced to relate a dimensionless parameter characterizing the deformation with the natural length of the harmonic oscillator. Contraction from q-deformed oscillators to c-deformed oscillators is accomplished by keeping lq finite while taking the limit ℏ → 0. The c-deformed Hamilton functions for both types of oscillators are found to be invariant under discrete translations: the step of the translation for the Dubna oscillator is half of that for the Macfarlane oscillator. The c-deformed oscillator of the Macfarlane type has propagating solutions in addition to localized ones. Reinvestigation of the g-deformed oscillator carried out in the light of these findings for the c-deformed systems proves that the g-deformed systems are invariant under the same translation symmetries as the c-deformed systems and have propagating waves of the Bloch typeArticle Citation - WoS: 15Citation - Scopus: 13Black Holes and Solitons of the Quantized Dispersionless Nls and Dnls Equations(Cambridge University Press, 2002) Pashaev, Oktay; Lee, Jyh HaoThe classical dynamics of non-relativistic particles are described by the Schrödinger wave equation, perturbed by quantum potential nonlinearity. Quantization of this dispersionless equation, implemented by deformation of the potential strength, recovers the standard Schrödinger equation. In addition, the classically forbidden region corresponds to the Planck constant analytically continued to pure imaginary, values. We apply the same procedure to the NLS and DNLS equations, constructing first the corresponding dispersionless limits and then adding quantum deformations. All these deformations admit the Lax representation as well as the Hirota bilinear form. In the classically forbidden region we find soliton resonances and black hole phenomena. For deformed DNLS the chiral solitons with single event horizon and resonance dynamics are constructed.Conference Object Citation - WoS: 6Citation - Scopus: 6Soliton Resonances, Black Holes and Madelung Fluid(Taylor and Francis Ltd., 2001) Pashaev, Oktay; Lee, Jyh HaoThe reaction-diffusion system realizing a particular gauge fixing condition of the Jackiw-Teitelboim gravity is represented as a coupled pair of Burgers equations with positive and negative viscosity. For acoustic metric in the Madelung fluid representation the space-time points where dispersion change the sign correspond to the event horizon, while shock soliton solutions to the black holes, creating under collision the resonance states.Conference Object Citation - WoS: 2Citation - Scopus: 2Self-Dual Chern-Simons Solitons and Quantum Potential(Taylor and Francis Ltd., 2001) Pashaev, Oktay; Lee, Jyh HaoAn influence of the quantum potential on the Chern-Simons solitons leads to quantization of the statistical parameter κ = me 2/g, and the quantum potential strength s = 1 - m 2. A new type of exponentially localized Chern-Simons solitons for the Bloch electrons near the hyperbolic energy band boundary are found.Article Citation - WoS: 17Citation - Scopus: 17Self-Dual Vortices in Chern-Simons Hydrodynamics(Pleiades Publishing, 2001) Lee, Jyh Hao; Pashaev, OktayThe classical theory of a nonrelativistic charged particle interacting with a U(1) gauge field is reformulated as the Schrödinger wave equation modified by the de Broglie-Bohm nonlinear quantum potential. The model is gauge equivalent to the standard Schrödinger equation with the Planck constant ℏ for the deformed strength 1 - ℏ2 of the quantum potential and to the pair of diffusion-antidiffusion equations for the strength 1 + ℏ2. Specifying the gauge field as the Abelian Chern-Simons (CS) one in 2+1 dimensions interacting with the nonlinear Schrödinger (NLS) field (the Jackiw-Pi model), we represent the theory as a planar Madelung fluid, where the CS Gauss law has the simple physical meaning of creation of the local vorticity for the fluid flow. For the static flow when the velocity of the center-of-mass motion (the classical velocity) is equal to the quantum velocity (generated by the quantum potential velocity of the internal motion), the fluid admits an N-vortex solution. Applying a gauge transformation of the Auberson-Sabatier type to the phase of the vortex wave function, we show that deformation parameter ℏ, the CS coupling constant, and the quantum potential strength are quantized. We discuss reductions of the model to 1+1 dimensions leading to modified NLS and DNLS equations with resonance soliton interactions.