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\ $u_{t}=F(t,x,u,u_{x})u_{xx} + G(t,x,u,u_{x})$. 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 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 $A(t, \vec x)=(A_0(t, \vec x)$, $\vec A(t, \vec x))$ 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

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 $n$ that admit an $N$-shock type solution with $N\leq n+1$. 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 $t$, 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

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 $u_t=F(t,x,u,u_x, u_{xx})u_{xxx}+G(t,x,u,u_x, u_{xx})$ 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 $\to$ CO$_2$ 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