Mathematics / Matematik

Permanent URI for this collectionhttps://hdl.handle.net/11147/8

Browse

Filters

2001 - 200922010 - 20112

Settings

Subject: search.filters.subject.Schrödinger equationWoS Q: search.filters.wosQuartile.Q3

Search Results

Now showing 1 - 4 of 4
  • Article
    Citation - WoS: 8
    Citation - Scopus: 8
    Some Exact and Explicit Solutions To a Two-Component, Discrete, Nonlinear Schrödinger Model
    (National Research Council of Canada, 2011) Aslan, İsmail
    Natural processes and phenomena often display discrete structure. The discrete nonlinear Schrödinger equations are used in both physics and biology to model periodic optical structures and energy transfer in proteins. In this study, we present a new application of the (G'/G)-expansion method to special, coupled, discrete, nonlinear Schrödinger-type equations. This application is shown to be an effective tool for constructing solitary and periodic wave profiles with arbitrary parameters. In addition, we provide rational solutions that have not been explicitly computed before.
  • Article
    Citation - WoS: 5
    Citation - Scopus: 5
    Madelung Representation of Damped Parametric Quantum Oscillator and Exactly Solvable Schrödinger-Burgers Equations
    (American Institute of Physics, 2010) Atılgan Büyükaşık, Şirin; Pashaev, Oktay
    We construct a Madelung fluid model with time variable parameters as a dissipative quantum fluid and linearize it in terms of Schrödinger equation with time-dependent parameters. It allows us to find exact solutions of the nonlinear Madelung system in terms of solutions of the Schrödinger equation and the corresponding classical linear ordinary differential equation with variable frequency and damping. For the complex velocity field, the Madelung system takes the form of a nonlinear complex Schrödinger-Burgers equation, for which we obtain exact solutions using complex Cole-Hopf transformation. In particular, we give exact results for nonlinear Madelung systems related with Caldirola-Kanai-type dissipative harmonic oscillator. Collapse of the wave function in dissipative models and possible implications for the quantum cosmology are discussed. © 2010 American Institute of Physics.
  • Article
    Citation - WoS: 15
    Citation - Scopus: 13
    Black Holes and Solitons of the Quantized Dispersionless Nls and Dnls Equations
    (Cambridge University Press, 2002) Pashaev, Oktay; Lee, Jyh Hao
    The 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.
  • Article
    Citation - WoS: 17
    Citation - Scopus: 17
    Self-Dual Vortices in Chern-Simons Hydrodynamics
    (Pleiades Publishing, 2001) Lee, Jyh Hao; Pashaev, Oktay
    The 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.