107 research outputs found
Group classification of (1+1)-Dimensional Schr\"odinger Equations with Potentials and Power Nonlinearities
We perform the complete group classification in the class of nonlinear
Schr\"odinger equations of the form
where is an arbitrary
complex-valued potential depending on and is a real non-zero
constant. We construct all the possible inequivalent potentials for which these
equations have non-trivial Lie symmetries using a combination of algebraic and
compatibility methods. The proposed approach can be applied to solving group
classification problems for a number of important classes of differential
equations arising in mathematical physics.Comment: 10 page
Group classification of heat conductivity equations with a nonlinear source
We suggest a systematic procedure for classifying partial differential
equations invariant with respect to low dimensional Lie algebras. This
procedure is a proper synthesis of the infinitesimal Lie's method, technique of
equivalence transformations and theory of classification of abstract low
dimensional Lie algebras. As an application, we consider the problem of
classifying heat conductivity equations in one variable with nonlinear
convection and source terms. We have derived a complete classification of
nonlinear equations of this type admitting nontrivial symmetry. It is shown
that there are three, seven, twenty eight and twelve inequivalent classes of
partial differential equations of the considered type that are invariant under
the one-, two-, three- and four-dimensional Lie algebras, correspondingly.
Furthermore, we prove that any partial differential equation belonging to the
class under study and admitting symmetry group of the dimension higher than
four is locally equivalent to a linear equation. This classification is
compared to existing group classifications of nonlinear heat conductivity
equations and one of the conclusions is that all of them can be obtained within
the framework of our approach. Furthermore, a number of new invariant equations
are constructed which have rich symmetry properties and, therefore, may be used
for mathematical modeling of, say, nonlinear heat transfer processes.Comment: LaTeX, 51 page
Equivalence of conservation laws and equivalence of potential systems
We study conservation laws and potential symmetries of (systems of)
differential equations applying equivalence relations generated by point
transformations between the equations. A Fokker-Planck equation and the Burgers
equation are considered as examples. Using reducibility of them to the
one-dimensional linear heat equation, we construct complete hierarchies of
local and potential conservation laws for them and describe, in some sense, all
their potential symmetries. Known results on the subject are interpreted in the
proposed framework. This paper is an extended comment on the paper of J.-q. Mei
and H.-q. Zhang [Internat. J. Theoret. Phys., 2006, in press].Comment: 10 page
New results on group classification of nonlinear diffusion-convection equations
Using a new method and additional (conditional and partial) equivalence
transformations, we performed group classification in a class of variable
coefficient -dimensional nonlinear diffusion-convection equations of the
general form We obtain new interesting cases of
such equations with the density localized in space, which have large
invariance algebra. Exact solutions of these equations are constructed. We also
consider the problem of investigation of the possible local trasformations for
an arbitrary pair of equations from the class under consideration, i.e. of
describing all the possible partial equivalence transformations in this class.Comment: LaTeX2e, 19 page
Group analysis and exact solutions of a class of variable coefficient nonlinear telegraph equations
A complete group classification of a class of variable coefficient
(1+1)-dimensional telegraph equations , is
given, by using a compatibility method and additional equivalence
transformations. A number of new interesting nonlinear invariant models which
have non-trivial invariance algebras are obtained. Furthermore, the possible
additional equivalence transformations between equations from the class under
consideration are investigated. Exact solutions of special forms of these
equations are also constructed via classical Lie method and generalized
conditional transformations. Local conservation laws with characteristics of
order 0 of the class under consideration are classified with respect to the
group of equivalence transformations.Comment: 23 page
Nonlocal symmetries of integrable two-field divergent evolutionary systems
Nonlocal symmetries for exactly integrable two-field evolutionary systems of
the third order have been computed. Differentiation of the nonlocal symmetries
with respect to spatial variable gives a few nonevolutionary systems for each
evolutionary system. Zero curvature representations for some new nonevolution
systems are presented
A Review on Aerosol-Based Direct-Write and Its Applications for Microelectronics
Aerosol-based direct-write refers to the additive process of printing CAD/CAM features from an apparatus which creates a liquid or solid aerosol beam. Direct-write technologies are poised to become useful tools in the microelectronics industry for rapid prototyping of components such as interconnects, sensors, and thin film transistors (TFTs), with new applications for aerosol direct-write being rapidly conceived. This paper aims to review direct-write technologies, with an emphasis on aerosol-based systems. The different currently available state-of-the-art systems such as Aerosol Jet CAB-DW, MCS, and aerodynamic lenses are described. A review and analysis of the physics behind the fluid-particle interactions including Stokes and Saffman force, experimental observations, and how a full understanding of theory and experiments can lead to new technology are presented. Finally, the applications of aerosol direct-write for microelectronics are discussed
Group Analysis of Variable Coefficient Diffusion-Convection Equations. I. Enhanced Group Classification
We discuss the classical statement of group classification problem and some
its extensions in the general case. After that, we carry out the complete
extended group classification for a class of (1+1)-dimensional nonlinear
diffusion--convection equations with coefficients depending on the space
variable. At first, we construct the usual equivalence group and the extended
one including transformations which are nonlocal with respect to arbitrary
elements. The extended equivalence group has interesting structure since it
contains a non-trivial subgroup of non-local gauge equivalence transformations.
The complete group classification of the class under consideration is carried
out with respect to the extended equivalence group and with respect to the set
of all point transformations. Usage of extended equivalence and correct choice
of gauges of arbitrary elements play the major role for simple and clear
formulation of the final results. The set of admissible transformations of this
class is preliminary investigated.Comment: 25 page
Investigation of transition frequencies of two acoustically coupled bubbles using a direct numerical simulation technique
The theoretical results regarding the ``transition frequencies'' of two
acoustically interacting bubbles have been verified numerically. The theory
provided by Ida [Phys. Lett. A 297 (2002) 210] predicted the existence of three
transition frequencies per bubble, each of which has the phase difference of
between a bubble's pulsation and the external sound field, while
previous theories predicted only two natural frequencies which cause such phase
shifts. Namely, two of the three transition frequencies correspond to the
natural frequencies, while the remaining does not. In a subsequent paper [M.
Ida, Phys. Rev. E 67 (2003) 056617], it was shown theoretically that transition
frequencies other than the natural frequencies may cause the sign reversal of
the secondary Bjerknes force acting between pulsating bubbles. In the present
study, we employ a direct numerical simulation technique that uses the
compressible Navier-Stokes equations with a surface-tension term as the
governing equations to investigate the transition frequencies of two coupled
bubbles by observing their pulsation amplitudes and directions of translational
motion, both of which change as the driving frequency changes. The numerical
results reproduce the recent theoretical predictions, validating the existence
of the transition frequencies not corresponding to the natural frequency.Comment: 18 pages, 8 figures, in pres
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