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The unipolar and bipolar macroscopic quantum models derived recently for instance in the area of charge transport are considered in spatial one-dimensional whole space in the present paper. These models consist of nonlinear fourth-order parabolic equation for unipolar case or coupled nonlinear fourth-order parabolic system for bipolar case. We show for the first time the self-similarity property of the macroscopic quantum models in large time. Namely, we show that there exists a unique global strong solution with strictly positive density to the initial value problem of the macroscopic quantum models which tends to a self-similar wave (which is not the exact solution of the models) in large time at an algebraic time-decay rate.
We introduce the notion of perturbations of quantum stochastic models using the series product, and establish the asymptotic convergence of sequences of quantum stochastic models under the assumption that they are related via a right series product p
The method of self-similar factor approximants is completed by defining the approximants of odd orders, constructed from the power series with the largest term of an odd power. It is shown that the method provides good approximations for transcendent
We consider semigroups ${alpha_t: ; tgeq 0}$ of normal, unital, completely positive maps $alpha_t$ on a von Neumann algebra ${mathcal M}$. The (predual) semigroup $ u_t (rho):= rho circ alpha_t$ on normal states $rho$ of $mathcal M$ leaves invariant
Starting from a particle model describing self-propelled particles interacting through nematic alignment, we derive a macroscopic model for the particle density and mean direction of motion. We first propose a mean-field kinetic model of the particle
Self-similar solutions of the so called Airy equations, equivalent to the dispersionless nonlinear Schrodinger equation written in Madelung coordinates, are found and studied from the point of view of complete integrability and of their role in the r