1,368 research outputs found
Nonlinear theory of quantum Brownian motion
A nonlinear theory of quantum Brownian motion in classical environment is
developed based on a thermodynamically enhanced nonlinear Schrodinger equation.
The latter is transformed via the Madelung transformation into a nonlinear
quantum Smoluchowski-like equation, which is proven to reproduce key results
from quantum and classical physics. The application of the theory to a free
quantum Brownian particle results in a nonlinear dependence of the position
dispersion on time, being quantum generalization of the Einstein law of
Brownian motion. It is shown that the time of decoherence for the transition
from quantum to classical diffusion is proportional to the square of the
thermal de Broglie wavelength divided by the Einstein diffusion constant
Construction of type II blow-up solutions for the energy-critical wave equation in dimension 5
We consider the semilinear wave equation with focusing energy-critical
nonlinearity in space dimension 5 with radial data. It is known that a solution
which blows up at in a neighborhood (in the energy
norm) of the family of solitons , asymptotically decomposes in the
energy space as a sum of a bubble and an asymptotic profile
, where and . We construct a blow-up solution of this type such that
is any pair of sufficiently regular functions with . For these solutions the concentration rate is . We
also provide examples of solutions with concentration rate for , related to the behaviour of the asymptotic profile
near the origin.Comment: 39 pages; the new version takes into account the remarks of the
referee
Quadratic Mean Field Games
Mean field games were introduced independently by J-M. Lasry and P-L. Lions,
and by M. Huang, R.P. Malham\'e and P. E. Caines, in order to bring a new
approach to optimization problems with a large number of interacting agents.
The description of such models split in two parts, one describing the evolution
of the density of players in some parameter space, the other the value of a
cost functional each player tries to minimize for himself, anticipating on the
rational behavior of the others.
Quadratic Mean Field Games form a particular class among these systems, in
which the dynamics of each player is governed by a controlled Langevin equation
with an associated cost functional quadratic in the control parameter. In such
cases, there exists a deep relationship with the non-linear Schr\"odinger
equation in imaginary time, connexion which lead to effective approximation
schemes as well as a better understanding of the behavior of Mean Field Games.
The aim of this paper is to serve as an introduction to Quadratic Mean Field
Games and their connexion with the non-linear Schr\"odinger equation, providing
to physicists a good entry point into this new and exciting field.Comment: 62 pages, 4 figure
Regularized degenerate multi-solitons
We report complex PT-symmetric multi-soliton solutions to the Korteweg de-Vries equation that asymptotically contain one-soliton solutions, with each of them possessing the same amount of finite real energy. We demonstrate how these solutions originate from degenerate energy solutions of the Schrödinger equation. Technically this is achieved by the application of Darboux-Crum transformations involving Jordan states with suitable regularizing shifts. Alternatively they may be constructed from a limiting process within the context Hirota’s direct method or on a nonlinear superposition obtained from multiple Bäcklund transformations. The proposed procedure is completely generic and also applicable to other types of nonlinear integrable systems
- …