Quantum lattice gas representation of some classical solitons

Department of Physics, William & Mary, Williamsburg, VA 23187, USA; Air Force Research Laboratory, 29 Randolph Road, Hanscom AFB, MA 01731, USA; Department of Electrical & Computer Engineering, Old Dominion University, Norfolk, VA 23529, USA
Physics Letters A (Impact Factor: 1.63). 01/2003; DOI: 10.1016/S0375-9601(03)00334-7

ABSTRACT A quantum lattice gas representation is determined for both the non-linear Schrödinger (NLS) and Korteweg–de Vries (KdV) equations. There is excellent agreement with the solutions from these representations to the exact soliton–soliton collisions of the integrable NLS and KdV equations. These algorithms could, in principle, be simulated on a hybrid quantum-classical computer.

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    ABSTRACT: The dynamics of vortex solitons is studied in a BEC superfluid. A quantum lattice-gas algorithm (measurementbased quantum computation) is employed to examine the dynamical behavior vortex soliton solutions of the Gross-Pitaevskii equation (phi4 interaction nonlinear Schroedinger equation). Quantum turbulence is studied in large grid numerical simulations: Kolmogorov spectrum associated with a Richardson energy cascade occurs on large flow scales. At intermediate scales, a new k-5.9 power law emerges, due to vortex filamentary reconnections associated with Kelvin wave instabilities (vortex twisting) coupling to sound modes and the exchange of intermediate vortex rings. Finally, at very small spatial scales a k-3 power law emerges, characterizing fluid dynamics occurring within the scale size of the vortex cores themselves. Poincaré recurrence is studied: in the free non-interacting system, a fast Poincaré recurrence occurs for regular arrays of line vortices. The recurrence period is used to demarcate dynamics driving a nonlinear quantum fluid towards turbulence, since fast recurrence is an approximate symmetry of the nonlinear quantum fluid at early times. This class of quantum algorithms is useful for studying BEC superfluid dynamics and, without modification, should allow for higher resolution simulations (with many components) on future quantum computers.
    Proc SPIE 05/2009;
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    ABSTRACT: The derivation,of the quantum,lattice Boltzmann,model,is reviewed,with special emphasis on recent developments of the model, namely, the extension to a multi-dimensional formulation,and the application to the computation,of the ground state of the Gross-Pitaevskii equation,(GPE). Numerical,results for the linear and non- linear Schr¨odinger,equation,and,for the ground,state solution of the GPE are also presented,and validated,against analytical results or other classical schemes,such as Crank-Nicholson. PACS: 02.70.-c, 03.65-w, 03.67.Lx Key words: Quantum lattice Boltzmann, multi-dimensions, imaginary-time model, linear and
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    ABSTRACT: In this paper, a higher-order accuracy lattice Boltzmann model for the complex Ginzburg-Landau equation is proposed. In order to obtain higher-order accuracy of truncation error and to overcome the drawbacks of “error rebound” in the previous models, a new assumption of additional distribution is presented to improve the accuracy of the model for the complex partial differential equation with nonlinear source term. As results, the complex Ginzburg-Landau equation is recovered with the fourth-order accuracy of truncation error. Based on this model, the problems of a single spiral wave in two-dimensional (2D) space and a single scroll in three-dimensional (3D) space are implemented to test the lattice Boltzmann scheme. The comparisons between the LBM results and the Alternative Direction Implicit results are given in detail. The numerical examples show that assumptions of source term can be used to raise the accuracy of the truncation error of the lattice Boltzmann scheme for the complex Ginzburg-Landau equation.
    Journal of Scientific Computing 01/2012; 52(3). · 1.71 Impact Factor

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