1,103 research outputs found
The Structure of Lie Algebras and the Classification Problem for Partial Differential Equations
The present paper solves completely the problem of the group classification
of nonlinear heat-conductivity equations of the form\
. We have proved, in particular,
that the above class contains no nonlinear equations whose invariance algebra
has dimension more than five. Furthermore, we have proved that there are two,
thirty-four, thirty-five, and six inequivalent equations admitting one-, two-,
three-, four- and five-dimensional Lie algebras, respectively. Since the
procedure which we use, relies heavily upon the theory of abstract Lie algebras
of low dimension, we give a detailed account of the necessary facts. This
material is dispersed in the literature and is not fully available in English.
After this algebraic part we give a detailed description of the method and then
we derive the forms of inequivalent invariant evolution equations, and compute
the corresponding maximal symmetry algebras. The list of invariant equations
obtained in this way contains (up to a local change of variables) all the
previously-known invariant evolution equations belonging to the class of
partial differential equations under study.Comment: 45 page
On the classification of conditionally integrable evolution systems in (1+1) dimensions
We generalize earlier results of Fokas and Liu and find all locally analytic
(1+1)-dimensional evolution equations of order that admit an -shock type
solution with .
To this end we develop a refinement of the technique from our earlier work
(A. Sergyeyev, J. Phys. A: Math. Gen, 35 (2002), 7653--7660), where we
completely characterized all (1+1)-dimensional evolution systems
\bi{u}_t=\bi{F}(x,t,\bi{u},\p\bi{u}/\p x,...,\p^n\bi{u}/\p x^n) that are
conditionally invariant under a given generalized (Lie--B\"acklund) vector
field \bi{Q}(x,t,\bi{u},\p\bi{u}/\p x,...,\p^k\bi{u}/\p x^k)\p/\p\bi{u} under
the assumption that the system of ODEs \bi{Q}=0 is totally nondegenerate.
Every such conditionally invariant evolution system admits a reduction to a
system of ODEs in , thus being a nonlinear counterpart to quasi-exactly
solvable models in quantum mechanics.
Keywords: Exact solutions, nonlinear evolution equations, conditional
integrability, generalized symmetries, reduction, generalized conditional
symmetries
MSC 2000: 35A30, 35G25, 81U15, 35N10, 37K35, 58J70, 58J72, 34A34Comment: 8 pages, LaTeX 2e, now uses hyperre
On separable Schr\"odinger equations
We classify (1+3)-dimensional Schr\"odinger equations for a particle
interacting with the electromagnetic field that are solvable by the method of
separation of variables. As a result, we get eleven classes of the
electromagnetic vector potentials of the electromagnetic field , providing separability of the
corresponding Schr\"odinger equations. It is established, in particular, that
the necessary condition for the Schr\"odinger equation to be separable is that
the magnetic field must be independent of the spatial variables. Next, we prove
that any Schr\"odinger equation admitting variable separation into second-order
ordinary differential equations can be reduced to one of the eleven separable
Schr\"odinger equations mentioned above and carry out variable separation in
the latter. Furthermore, we apply the results obtained for separating variables
in the Hamilton-Jacobi equation.Comment: 30 pages, LaTe
Catalytic CO Oxidation on Nanoscale Pt Facets: Effect of Inter-Facet CO Diffusion on Bifurcation and Fluctuation Behavior
We present lattice-gas modeling of the steady-state behavior in CO oxidation
on the facets of nanoscale metal clusters, with coupling via inter-facet CO
diffusion. The model incorporates the key aspects of reaction process, such as
rapid CO mobility within each facet, and strong nearest-neighbor repulsion
between adsorbed O. The former justifies our use a "hybrid" simulation approach
treating the CO coverage as a mean-field parameter. For an isolated facet,
there is one bistable region where the system can exist in either a reactive
state (with high oxygen coverage) or a (nearly CO-poisoned) inactive state.
Diffusion between two facets is shown to induce complex multistability in the
steady states of the system. The bifurcation diagram exhibits two regions with
bistabilities due to the difference between adsorption properties of the
facets. We explore the role of enhanced fluctuations in the proximity of a cusp
bifurcation point associated with one facet in producing transitions between
stable states on that facet, as well as their influence on fluctuations on the
other facet. The results are expected to shed more light on the reaction
kinetics for supported catalysts.Comment: 22 pages, RevTeX, to appear in Phys. Rev. E, 6 figures (eps format)
are available at http://www.physik.tu-muenchen.de/~natali
Symmetry classification of third-order nonlinear evolution equations. Part I: Semi-simple algebras
We give a complete point-symmetry classification of all third-order evolution
equations of the form
which admit semi-simple symmetry algebras and extensions of these semi-simple
Lie algebras by solvable Lie algebras. The methods we employ are extensions and
refinements of previous techniques which have been used in such
classifications.Comment: 53 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
Wigner phase space distribution as a wave function
We demonstrate that the Wigner function of a pure quantum state is a wave
function in a specially tuned Dirac bra-ket formalism and argue that the Wigner
function is in fact a probability amplitude for the quantum particle to be at a
certain point of the classical phase space. Additionally, we establish that in
the classical limit, the Wigner function transforms into a classical
Koopman-von Neumann wave function rather than into a classical probability
distribution. Since probability amplitude need not be positive, our findings
provide an alternative outlook on the Wigner function's negativity.Comment: 6 pages and 2 figure
Decay of metastable phases in a model for the catalytic oxidation of CO
We study by kinetic Monte Carlo simulations the dynamic behavior of a
Ziff-Gulari-Barshad model with CO desorption for the reaction CO + O
CO on a catalytic surface. Finite-size scaling analysis of the fluctuations
and the fourth-order order-parameter cumulant show that below a critical CO
desorption rate, the model exhibits a nonequilibrium first-order phase
transition between low and high CO coverage phases. We calculate several points
on the coexistence curve. We also measure the metastable lifetimes associated
with the transition from the low CO coverage phase to the high CO coverage
phase, and {\it vice versa}. Our results indicate that the transition process
follows a mechanism very similar to the decay of metastable phases associated
with {\it equilibrium} first-order phase transitions and can be described by
the classic Kolmogorov-Johnson-Mehl-Avrami theory of phase transformation by
nucleation and growth. In the present case, the desorption parameter plays the
role of temperature, and the distance to the coexistence curve plays the role
of an external field or supersaturation. We identify two distinct regimes,
depending on whether the system is far from or close to the coexistence curve,
in which the statistical properties and the system-size dependence of the
lifetimes are different, corresponding to multidroplet or single-droplet decay,
respectively. The crossover between the two regimes approaches the coexistence
curve logarithmically with system size, analogous to the behavior of the
crossover between multidroplet and single-droplet metastable decay near an
equilibrium first-order phase transition.Comment: 27 pages, 22 figures, accepted by Physical Review
ЛАВИННЫЙ РИСК В КАЗАХСТАНЕ ПРИ РАЗЛИЧНЫХ УРОВНЯХ ЛАВИННОЙ ОПАСНОСТИ
A review was made of avalanche accidents in Kazakhstan. Data on casualties and damage processed for the period 1951–2020. The aim of the study is to analyze avalanche incidents and develop recommendations for their prevention. The relationship between accidents and the degree of avalanche danger in mountainous areas is also considered. In total, over the period studied, 95 accidents occurred in Kazakhstan, 95 people died and 93 people were injured. The largest number of avalanche incidents occurred in the vicinity of the city of Almaty on the territory of the Ile-Alatau National Park – 81%. The most avalanche months are March-April (38 and 21%, respectively). The main victims of avalanches are extreme sports enthusiasts. Most of the victims themselves provoked avalanches – 56% of cases.
The relationship between the level of avalanche danger and avalanche risk was studied. For this, data were collected on avalanche situations in the area of the Shymbulak avalanche station for the observation period 1978–2020. The classification of the level of avalanche danger was carried out in accordance with an international five-point scale. In the cold period from November to May, the second "yellow" hazard level prevails here – 56% of days. The extreme level of danger was noted only in 0.2% of days. Avalanche incidents with victims mainly occurred at the first and second levels of avalanche danger: 21 and 52% of cases. Material damage to objects was caused during the descent of catastrophic avalanches at the fourth and fifth levels of avalanche danger (43 and 29% of cases, respectively).
Analysis of avalanche incidents allows us to conclude that the majority of avalanche victims are extreme sports enthusiasts who die in provoked avalanches at low levels of avalanche danger. To prevent such accidents, it is necessary, instead of "Storm warnings", to switch to probabilistic forecasts using an international five-point danger scale. At low levels of danger, it is very important to carry out preventive measures among fans of extreme sports – lectures, seminars and training courses.Авторами проведен обзор несчастных случаев, связанных со снежными лавинами в Казахстане. Обработаны данные о жертвах и ущербе за период с 1951 по 2020 год. Целью исследования является анализ лавинных инцидентов и разработка рекомендаций по их предотвращению. Также рассмотрена связь несчастных случаев со степенью лавинной опасности в горных районах.
Всего за изученный период в Казахстане произошло 95 несчастных случаев, погибло 95 и пострадало 93 человека. Наибольшее количество лавинных инцидентов произошло в окрестностях города Алматы на территории Иле-Алатауского национального парка – 81%. Самые лавиноопасные месяцы: март – апрель (38 и 21% соответственно). Основные жертвы снежных лавин – любители экстремальных видов спорта. Большинство жертв сами спровоцировали сход лавин – 56% случаев.
Была изучена связь между уровнем лавинной опасности и лавинным риском. Для этого собраны данные о лавиноопасных ситуациях в районе снеголавинной станции «Шымбулак» за период наблюдений с 1978 по 2020 год. Классификация уровня лавинной опасности была проведена в соответствии с международной пятибалльной шкалой. В холодный период с ноября по май преобладает второй «желтый» уровень опасности – 56% дней. Экстремальный уровень опасности отмечался только в 0,2% дней. Лавинные инциденты с жертвами преимущественно происходили при первом и втором уровнях лавинной опасности: 21 и 52% случаев. Материальный ущерб отмечался при четвертом и пятом уровнях лавинной опасности (43 и 29% случаев соответственно).
Анализ лавинных инцидентов позволяет сделать вывод: большинство жертв лавин – это любители экстремальных видов спорта, которые погибают в спровоцированных лавинах при низких уровнях лавинной опасности. Для профилактики таких несчастных случаев необходимо переходить от «Штормовых предупреждений» к вероятностным прогнозам с применением международной пятибалльной шкалы опасности. При низких уровнях опасности очень важно проводить профилактические мероприятия среди любителей экстремального спорта – лекции, семинары, обучающие курсы
Литература:
Благовещенский В.П., Жданов В.В. Опыт оценки и прогноза лавинной опасности в Швейцарии // Гидрометеорология и экология. 2019. № 1 (92). С. 178–191.
Жданов В.В. Анализ несчастных случаев, связанных с лавинами // Вопросы географии и геоэкологии. 2012. № 2. С. 26–30.
Жданов В.В. Анализ ошибок снеголавинных наблюдений и прогнозов // Вопросы географии и геоэкологии. 2015. № 3. С. 52–55.
Жданов В.В. Обзор несчастных случаев в Казахстанском альпинизме в 2004–2019 годах // Вопросы географии и геоэкологии. 2019. № 4. С. 73–79.
Медеу А.Р., Благовещенский В.П., Жданов В.В. Инновационные технологии оценки и прогноза уровня лавинной опасности в горах Иле Алатау // Вестник КазНУ. Серия Географическая. 2021. №2 (61) 2021. С. 76–87. DOI: 10.26577/JGEM.2021.v61.i2.07.
Практическое пособие по прогнозированию лавинной опасности в Казахстане / Сост. Е.И. Колесников. Алматы: РГП «Казгидромет», 2005. 262 с.
Greene E., Birkeland K., Elder K., McCammon I., Staples M., Sharaf D. Snow, weather, and avalanches: Observation guidelines for avalanche programs in the United States. Victor: American Avalanche Association, 2016. 104 p.
Observation Guidelines and Recording Standards for Weather, Snowpack and Avalanches. Revelstoke: Canadian Avalanche Association, 2014. 109 p.
Schweizer J., Mitterer C., Techel1 F., Stoffel1 A., Reuter B. On the relation between avalanche occurrence and avalanche danger level / The Cryosphere. 2020. Vol. 14. Iss. 2. Р. 737–750. DOI: 10.5194/tc-14-737-2020.
Techel F., Schweizer J. On using local avalanche danger level estimates for regional forecast verification // Cold Regions Science and Technology. 2017. Vol. 144. Рp. 52–62. DOI: 10.1016/j.coldregions.2017.07.012
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