Mathematics / Matematik
Permanent URI for this collectionhttps://hdl.handle.net/11147/8
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Article Citation - WoS: 4Citation - Scopus: 6Dissipative Hierarchies and Resonance Solitons for Kp-Ii and Mkp-Ii(Elsevier Ltd., 2007) Francisco, Meltem L. Y.; Lee, Jyh Hao; Pashaev, OktayWe show that dissipative solitons (dissipatons) of the second and the third members of SL(2,R) AKNS hierarchy give rise to the real solitons of KP-II, while for SL(2,R) Kaup-Newell hierarchy they give solitons of MKP-II. By the Hirota bilinear form for both flows, we find new bilinear system for these equations, and one- and two-soliton solutions. For special values of parameters our solutions show resonance behaviour with creation of four virtual solitons. We first time created four virtual soliton resonance solution for KP-II and established relations of it with degenerate four-soliton solution in the Hirota-Satsuma bilinear form for KP-II. Our approach allows one to interpret the resonance soliton as a composite object of two dissipative solitons in 1 + 1 dimensions.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.Conference Object Citation - WoS: 24Citation - Scopus: 24Solitons of the Resonant Nonlinear Schrödinger Equation With Nontrivial Boundary Conditions: Hirota Bilinear Method(Pleiades Publishing, 2007) Lee, Jyh Hao; Pashaev, OktayWe use the Hirota bilinear approach to consider physically relevant soliton solutions of the resonant nonlinear Schrödinger equation with nontrivial boundary conditions, recently proposed for describing uniaxial waves in a cold collisionless plasma. By the Madelung representation, the model transforms into the reaction-diffusion analogue of the nonlinear Schrödinger equation, for which we study the bilinear representation, the soliton solutions, and their mutual interactions.Article Citation - WoS: 43Citation - Scopus: 43Shock Waves, Chiral Solitons and Semiclassical Limit of One-Dimensional Anyons(Elsevier Ltd., 2004) Lee, Jyh Hao; Lin, Chi-Kun; Pashaev, OktayThis paper is devoted to the semiclassical limit of the one-dimensional Schrödinger equation with current nonlinearity and Sobolev regularity, before shocks appear in the limit system. In this limit, the modified Euler equations are recovered. The strictly hyperbolicity and genuine nonlinearity are proved for the limit system wherever the Riemann invariants remain distinct. The dispersionless equation and its deformation which is the quantum potential perturbation of JNLS equation are also derived.Article Citation - WoS: 91Citation - Scopus: 91Resonance Solitons as Black Holes in Madelung Fluid(World Scientific Publishing Co. Pte Ltd, 2002) Pashaev, Oktay; Lee, Jyh HaoEnvelope solitons of the Nonlinear Schrödinger equation (NLS) under quantum potential's influence are studied. Corresponding problem is found to be integrable for an arbitrary strength, s ≠ 1, of the quantum potential. For s < 1, the model is equivalent to the usual NLS with rescaled coupling constant, while for s > 1, to the reaction-diffusion system. The last one is related to the anti-de Sitter (AdS) space valued Heisenberg model, realizing a particular gauge fixing condition of the (1 + 1)-dimensional Jackiw-Teitelboim gravity. For this gravity model, by the Madelung fluid representation we derive the acoustic form of the space-time metric. The space-time points, where dispersion changes the sign, correspond to the event horizon, while the soliton solution to the AdS black hole. Moving with the above bounded velocity, it describes evolution on the one sheet hyperboloid with nontrivial winding number, and creates under collision, the resonance states which we study by the Hirota bilinear method.Article 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.