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Register a new userThe large nonlinearity scale limit of an information-theoretically motivated nonlinear Schrodinger equation
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A nonlinear Schrodinger equation, that had been obtained within the context of the maximum uncertainty principle, has the form of a difference-differential equation and exhibits some interesting properties. Here we discuss that equation in the regime where the nonlinearity length scale is large compared to the deBroglie wavelength; just as in the perturbative regime, the equation again displays some universality. We also briefly discuss stationary solutions to a naturally induced discretisation of that equation.
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I begin by reviewing the arguments leading to a nonlinear generalisation of Schrodingers equation within the context of the maximum uncertainty principle. Some exact and perturbative properties of that equation are then summarised: those results depend on a free regulating/interpolation parameter $eta$. I discuss here how one may fix that parameter using energetics. Other issues discussed are, a linear theory with an external potential that reproduces some unusual exact solutions of the nonlinear equation, and possible symmetry enhancements in the nonlinear theory.
We obtain novel nonlinear Schr{o}dinger-Pauli equations through a formal non-relativistic limit of appropriately constructed nonlinear Dirac equations. This procedure automatically provides a physical regularisation of potential singularities brought forward by the nonlinear terms and suggests how to regularise previous equations studied in the literature. The enhancement of contributions coming from the regularised singularities suggests that the obtained equations might be useful for future precision tests of quantum nonlinearity.
We update our understanding of nonlinear Schrodinger equations motivated through information theory. In particular we show that a $q-$deformation of the basic nonlinear equation leads to a perturbative increase in the energy of a system, thus favouring the simplest $q=1$ case. Furthermore the energy minimisation criterion is shown to be equivalent, at leading order, to an uncertainty maximisation argument. The special value $eta =1/4$ for the interpolation parameter, where leading order energy shifts vanish, implies the preservation of existing supersymmetry in nonlinearised supersymmetric quantum mechanics. Physically, $eta$ might be encoding relativistic effects.
In this paper, classical small perturbations against a stationary solution of the nonlinear Schrodinger equation with the general form of nonlinearity are examined. It is shown that in order to obtain correct (in particular, conserved over time) nonzero expressions for the basic integrals of motion of a perturbation even in the quadratic order in the expansion parameter, it is necessary to consider nonlinear equations of motion for the perturbations. It is also shown that, despite the nonlinearity of the perturbations, the additivity property is valid for the integrals of motion of different nonlinear modes forming the perturbation (at least up to the second order in the expansion parameter).
Utilization of a quantum system whose time-development is described by the nonlinear Schrodinger equation in the transformation of qubits would make it possible to construct quantum algorithms which would be useful in a large class of problems. An example of such a system for implementing the logical NOR operation is demonstrated.
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