19 research outputs found
The focusing logarithmic Schrödinger equation: analysis of breathers and nonlinear superposition
We consider the logarithmic Schrödinger equation in the focusing regime. For this equation, Gaussian initial data remains Gaussian. In particular, the Gausson - a time-independent Gaussian function - is an orbitally stable solution. In the general case in dimension d = 1, the solution with Gaussian initial data is periodic, and we compute some approximations of the period in the case of small and large oscillations, showing that the period can be as large as wanted for the latter. The main result of this article is a principle of nonlinear superposition: starting from an initial data made of the sum of several standing Gaussian functions far from each other, the solution remains close (in L^2) to the sum of the corresponding Gaussian solutions for a long time, in square of the distance between the Gaussian functions
Error analysis of a first-order IMEX scheme for the logarithmic Schr\"odinger equation
The logarithmic Schr\"odinger equation (LogSE) has a logarithmic nonlinearity
that is not differentiable at Compared with its
counterpart with a regular nonlinear term, it possesses richer and unusual
dynamics, though the low regularity of the nonlinearity brings about
significant challenges in both analysis and computation. Among very limited
numerical studies, the semi-implicit regularized method via regularising
as to overcome the
blowup of at has been investigated recently in literature.
With the understanding of we analyze the non-regularized first-order
Implicit-Explicit (IMEX) scheme for the LogSE. We introduce some new tools for
the error analysis that include the characterization of the H\"older continuity
of the logarithmic term, and a nonlinear Gr\"{o}nwall's inequality. We provide
ample numerical results to demonstrate the expected convergence. We position
this work as the first one to study the direct linearized scheme for the LogSE
as far as we can tell.Comment: 19 pages, 5 figure
Dividing Line between Quantum and Classical Trajectories: Bohmian Time Constant
This work proposes an answer to a challenge posed by Bell on the lack of
clarity in regards to the line between the quantum and classical regimes in a
measurement problem. To this end, a generalized logarithmic nonlinear
Schr\"odinger equation is proposed to describe the time evolution of a quantum
dissipative system under continuous measurement. Within the Bohmian mechanics
framework, a solution to this equation reveals a novel result: it displays a
time constant which should represent the dividing line between the quantum and
classical trajectories. It is shown that continuous measurements and damping
not only disturb the particle but compel the system to converge in time to a
Newtonian regime. While the width of the wave packet may reach a stationary
regime, its quantum trajectories converge exponentially in time to classical
trajectories. In particular, it is shown that damping tends to suppress further
quantum effects on a time scale shorter than the relaxation time of the system.
If the initial wave packet width is taken to be equal to 2.8 10^{-15} m (the
approximate size of an electron), the Bohmian time constant is found to have an
upper limit, i. e.,
The Cauchy problem for the logarithmic Schr\"odinger equation revisited
We revisit the Cauchy problem for the logarithmic Schr\"odinger equation and
construct strong solutions in , the energy space, and the -energy
space. The solutions are provided in a constructive way, which does not rely on
compactness arguments, that a sequence of approximate solutions forms a Cauchy
sequence in a complete function space and then actual convergence is shown to
be in a strong sense.Comment: 30 page
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