Wellposedness of a nonlinear, logarithmic Schrödinger equation of Doebner-Goldin type modeling quantum dissipation

This paper is concerned with the modeling and analysis of quantum dissipation phenomena in the Schrödinger picture. More precisely, we do investigate in detail a dissipative, nonlinear Schrödinger equation somehow accounting for quantum Fokker-Planck effects, and how it is drastically reduced to a simpler logarithmic equation via a nonlinear gauge transformation in such a way that the physics underlying both problems keeps unaltered. From a mathematical viewpoint, this allows for a more achievable analysis regarding the local wellposedness of the initial-boundary value problem. This simplification requires the performance of the polar (modulus-argument) decomposition of the wavefunction, which is rigorously attained (for the first time to the best of our knowledge) under quite reasonable assumptions

A repertoire of repulsive Keller–Segel models with logarithmic sensitivity: Derivation, traveling waves, and quasi-stationary dynamics

In this paper, we show how the chemotactic model {partial derivative(t)rho = d(1) Delta(x)rho - del(x) . (rho del(x)c) partial derivative(t)c = d(2) Delta(x)c + F(rho, c, del(x)rho, del(x)c, Delta x rho) introduced in Alejo and Lopez (2021), which accounts for a chemical production-degradation operator of Hamilton-Jacobi type involving first- and second-order derivatives of the logarithm of the cell concentration, namely, F = mu + tau c - sigma rho + A Delta(x)rho/rho + B vertical bar del(x)rho vertical bar(2)/rho(2) + C vertical bar del(x)c vertical bar(2), with mu, tau, sigma, A, B, C is an element of R, can be formally reduced to a repulsive Keller-Segel model with logarithmic sensitivity { partial derivative(t)rho = D-1 Delta(x)rho + chi del(x) . (rho del(x) log(c)), chi, lambda, beta > 0, partial derivative(t)c = D-2 Delta(x)c + lambda rho c - beta c whenever the chemotactic parameters are appropriately chosen and the cell concentration keeps strictly positive. In this way, some explicit solutions (namely, traveling waves and stationary cell density profiles) of the former system can be transferred to a number of variants of the the latter by means of an adequate change of variables.Spanish Government RTI2018-098850-B-I00 Junta de AndaluciaEuropean Commission PY18-RT-2422 B-FQM-580-UGRUniversidad de Granada/CBU

On a rigorous interpretation of the quantum Schrödinger–Langevin operator in bounded domains with applications

AbstractIn this paper we make it mathematically rigorous the formulation of the following quantum Schrödinger–Langevin nonlinear operator for the wavefunctionAQSL=iℏ∂t+ℏ22mΔx−λ(Sψ−〈Sψ〉)−Θℏ[nψ,Jψ] in bounded domains via its mild interpretation. The a priori ambiguity caused by the presence of the multi-valued potential λSψ, proportional to the argument of the complex-valued wavefunctionψ=|ψ|exp{iℏSψ}, is circumvented by subtracting its positional expectation value,〈Sψ〉(t):=∫ΩSψ(t,x)nψ(t,x)dx, as motivated in the original derivation (Kostin, 1972 [45]). The problem to be solved in order to find Sψ is mostly deduced from the modulus-argument decomposition of ψ and dealt with much like in Guerrero et al. (2010) [37]. Here ℏ is the (reduced) Planck constant, m is the particle mass, λ is a friction coefficient, nψ=|ψ|2 is the local probability density, Jψ=ℏmIm(ψ¯∇xψ) denotes the electric current density, and Θℏ is a general operator (eventually nonlinear) that only depends upon the macroscopic observables nψ and Jψ. In this framework, we show local well-posedness of the initial-boundary value problem associated with the Schrödinger–Langevin operator AQSL in bounded domains. In particular, all of our results apply to the analysis of the well-known Kostin equation derived in Kostin (1972) [45] and of the Schrödinger–Langevin equation with Poisson coupling and enthalpy dependence (Jüngel et al., 2002 [41])

Wellposedness of a nonlinear, logarithmic Schrödinger equation of Doebner-Goldin type modeling quantum dissipation

This paper is concerned with the modeling and analysis of quantum dissipation phenomena in the Schrödinger picture. More precisely, we do investigate in detail a dissipative, nonlinear Schrödinger equation somehow accounting for quantum Fokker–Planck effects, and how it is drastically reduced to a simpler logarithmic equation via a nonlinear gauge transformation in such a way that the physics underlying both problems keeps unaltered. From a mathematical viewpoint, this allows for a more achievable analysis regarding the local wellposedness of the initial–boundary value problem. This simplification requires the performance of the polar (modulus–argument) decomposition of the wavefunction, which is rigorously attained (for the first time to the best of our knowledge) under quite reasonable assumptions