1,188 research outputs found
Very Singular Similarity Solutions and Hermitian Spectral Theory for Semilinear Odd-Order PDEs
Very singular self-similar solutions of semilinear odd-order PDEs are studied
on the basis of a Hermitian-type spectral theory for linear rescaled odd-order
operators.Comment: 49 pages, 12 Figure
Statistics of Current Fluctuations and Electron-Electron Interactions in Mesoscopic Coherent Conductors
We formulate a general path integral approach which describes statistics of
current fluctuations in mesoscopic coherent conductors at arbitrary frequencies
and in the presence of interactions. Applying this approach to the
non-interacting case, we analyze the frequency dispersion of the third cumulant
of the current operator at frequencies well below both the inverse
charge relaxation time and the inverse electron dwell time. This dispersion
turns out to be important in the frequency range comparable to applied
voltages. For comparatively transparent conductors it may lead to the sign
change of . We also analyze the behavior of the second cumulant of
the current operator (current noise) in the presence of
electron-electron interactions. In a wide range of parameters we obtain
explicit universal dependencies of on temperature, voltage and
frequency. We demonstrate that Coulomb interaction decreases the Nyquist noise.
In this case the interaction correction to the noise spectrum is governed by
the combination , where is the transmission of the
-th conducting mode. The effect of electron-electron interactions on the
shot noise is more complicated. At sufficiently large voltages we recover two
different interaction corrections entering with opposite signs. The net result
is proportional to , i.e. Coulomb interaction
decreases the shot noise at low transmissions and increases it at high
transmissions.Comment: 12 pages, 3 figures. To be published in the Proceedings of the SPIE
Symposium on Fluctuations and Noise, Maspalomas, Grand Canaria, Spain (May
2004
Five types of blow-up in a semilinear fourth-order reaction-diffusion equation: an analytic-numerical approach
Five types of blow-up patterns that can occur for the 4th-order semilinear
parabolic equation of reaction-diffusion type
u_t= -\Delta^2 u + |u|^{p-1} u \quad {in} \quad \ren \times (0,T), p>1,
\quad \lim_{t \to T^-}\sup_{x \in \ren} |u(x,t)|= +\iy, are discussed. For
the semilinear heat equation , various blow-up patterns
were under scrutiny since 1980s, while the case of higher-order diffusion was
studied much less, regardless a wide range of its application.Comment: 41 pages, 27 figure
Ordinary differential equations which linearize on differentiation
In this short note we discuss ordinary differential equations which linearize
upon one (or more) differentiations. Although the subject is fairly elementary,
equations of this type arise naturally in the context of integrable systems.Comment: 9 page
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