This article discusses the theoretical and practical issues of hydroclimatic mapping. Hydroclimatic mapping is an important part of solving of environmental and water management problems, environmental management issues, and the adaptation of the population to climate change. Hydrological and climatic maps included in atlases of the Baikal region (Irkutsk Region: Ecological Conditions of Development (2004), Ecological Atlas of the Lake Baikal Basin (2015), and Baikal Region: Society and Nature (2021)) are presented. The authors consider the methodological approaches of hydroclimatic mapping. The limitations in using the traditional method of isolines are shown. The advantages of the basin method, landscape–hydroclimatic and structural–hydrographic approaches using zoning tools and along channel diagrams are substantiated. The article presents the methodology of flood-risk zoning, water protection, and recreational zoning of the Baikal coast and cartographic modeling of processes in the delta of the Selenga River.

General and particular solutions of the so called semi-Hamiltonian hydrodynamic type systems can be obtained by the Tsarev Generalized Hodograph Method. Here we show that a natural extension of this approach applied to dispersive integrable systems is determined by isomonodromic deformations.

The seismicity review of Kamchatka and surrounding territories for 2020 is given. In the Kam chatka earthquake catalogue, the minimum local magnitude of completeness is MLmin=3.5, and for the Kamchatka seismically active region (=50.5–56.5° N, =156.5–167° E) MLmin=3.0, and for earthquakes with h≥350 km under the Okhotsk sea MLmin=4.1. The Kamchatka earthquake catalogue for 2020, published in the Appendix to this article, includes 1666 events with ML≥3.5; 94 earthquakes with ML=3.55–7.65 were felt in Kamchatka and sur rounding areas with seismic intensity I of 1–2 to 6–7 according to the Seismic Intensity Scale-2017 (Russian state standard). For 49 events with ML≥5.0 that occurred in 2020 within the area of responsibility of Кamchatka branch of Geophysical Survey RAS, an attempt to calculate the seismic moment tensor (SMT) was made. The level of seismicity according to the "SOUS'09" scale in 2020 corresponded to the “high”. On March 25, 2020, there was a strong earthquake with Mw=7.4, named “the Paramushirskoe earthquake”. The earthquake was accompanied by a large number of aftershocks.

We consider compatible pairs of local Hamiltonian structures of Dubrovin–Novikov type and the construction of nonlocal Hamiltonian structures of Ferapontov type. We prove that the third and fourth Hamiltonian structures of Ferapontov type cannot contain more terms in the nonlocal part than the number of field variables. We also give local Lagrangian representations for both these nonlocal Hamiltonian structures. Thus, we demonstrate that any hydrodynamic-type system equipped with a compatible pair of local Hamiltonian structures of Dubrovin–Novikov type simultaneously possesses four local Lagrangian representations.

The article presents instrumental and macroseismic data on the Paramushir earthquake of March 25, 2020, ML=7.7, Mw=7.4, discusses its tectonic position and features of the aftershock process. This event is the strongest instrumentally recorded earthquake with a source located in the Pacific lithospheric plate in the area of the Northern Kuril Islands. The focal mechanisms and moment magnitude values Mw of the Paramushir earthquake and its strongest aftershocks were obtained using an original method for calculating seismic moment tensors, de veloped at the Кamchatka branch of Geophysical Survey RAS. The Paramushir earthquake was felt in 60 settle ments in the Kamchatka and the Sakhalin Regions, and was also noticed on the islands of Hokkaido (Japan) and Adak (USA). The maximum macroseismic manifestations were noted in the city of Severo-Kurilsk (Paramushir Island), I=6–7 points on the Seismic Intensity Scale 2017 (Russian building code GOST R 57546–2017); there were no casualties or destruction. A weak tsunami with a maximum observed wave height of ~50 cm was noted in the area of Severo-Kurilsk.

The seismicity review of Kamchatka and surrounding territories for 2018–2019 is given. In the Kamchatka earthquake catalogue, the minimum local magnitude of completeness is MLmin=3.8, and for the Kamchatka seismically active region (latitude = 50.5–56.5° N, longitude = 156.5–167° E) MLmin=3.7, and for earthquakes with h≥350 km under the Okhotsk sea MLmin=3.8. The Kamchatka earthquake catalogue for 2018–2019, published in the Appendix to this article, includes 3646 events with ML≥3.5; 228 earthquakes with ML=3.65–7.3 were felt in Kamchatka and surrounding areas with seismic intensity I of 1–2 to 6–7 according to the Seismic Intensity Scale-2017 (Russian state standard). For 134 events with ML≥5.0 that occurred in 2018–2019 within the area of responsibility of Кamchatka branch of Geophysical Survey RAS, an attempt to calculate the seismic moment tensor (SMT) was made. The SMT and depth h of the equivalent point source were calculated for 67 earthquakes in 2018 with a range of ML=5.0–7.3, and for 67 events in 2019 with a range of ML=5.0–6.45. The level of seismicity according to the "SOUS'09" scale in 2018 corresponded to the “background increased”, but within the assessment accuracy – “high”; for 2019 it was the “background average”. On December 20, 2018, there was a strong earthquake with Mw=7.3, named “the Angular Uplift earthquake”. This earthquake was the strongest intraplate event, which belongs to the region south of the junction zone of the Kamchatka and Aleutian trenches The earthquake was accompanied by a large number of aftershocks.

The strong Mw = 7.4 earthquake occurred on March 25, 2020 in the region of Northern Kuril Islands, with its epicenter on the ocean side of the Kuril-Kamchatka deep-water trench, to the east from its axis. The earthquake was felt on all the Kuril Islands, South and East Kamchatka, the maximum shaking was recorded in Severo-Kurilsk with intensity I = VI–VII. Distinct tsunami wave was also registered. In the article, this earthquake and its tectonic position are discussed in the context of the seismicity of the Kuril-Kamchatka arc. The actions of duty shifts in earthquake processing are described, and a detailed description of macroseismic effects is given. The results of the analysis of peak ground motion amplitudes, focal mechanisms and models of the source, tsunami propagation features are also shown. Coseismic displacements revealed by GNSS observation data are presented and compared with model data. Peculiarities and stages of the aftershock process are discussed and the size of the source area is estimated.

In this paper we continue investigation of a new property of two‐dimensional integrable systems—existence of infinitely many local three‐dimensional conservation laws for pairs of integrable two‐dimensional commuting flows. Multicomponent two‐dimensional hydrodynamic reductions of the Mikhalëv equation are considered. Infinitely many three‐dimensional local conservation laws for the Korteweg–de Vries pair of commuting flows are constructed. Thus, we show that pairs of commuting dispersive two‐dimensional systems also possess infinitely many local three‐dimensional conservation laws. They can be used for averaging of multiparametric families of solutions to the Mikhalëv equation.

Macroscopic dynamics of soliton gases can be analytically described by the thermodynamic limit of the Whitham equations, yielding an integro-differential kinetic equation for the density of states. Under a delta-functional ansatz, the kinetic equation for soliton gas reduces to a non-diagonalisable system of hydrodynamic type whose matrix consists of several 2x2 Jordan blocks. Here we demonstrate the integrability of this system by showing that it possesses a hierarchy of commuting hydrodynamic flows and can be solved by an extension of the generalised hodograph method. Our approach is a generalisation of Tsarev's theory of diagonalisable systems of hydrodynamic type to quasilinear systems with non-trivial Jordan block structure.

The seismicity review of Kamchatka and surrounding territories for 2016–2017 is given. In the Kamchatka earthquake catalogue, the minimum local magnitude of completeness is MLmin=4.0, and for earth-quakes with h≥350 km under the Okhotsk sea MLmin=4.3. The Kamchatka catalogue of earthquakes with ML=3.6–7.3 for 2016–2017, published in the Appendix to this article, includes 2898 events. 191 earthquakes of the catalogue were felt in Kamchatka and surrounding areas with seismic intensity I=2–6 according to the MSK-64 scale. For all events with ML≥5.0 that occurred in 2016–2017 within the area of responsibility of Kamchatka branch of Geophysical Survey RAS, an attempt to calculate the seismic moment tensor (SMT) was made. There are 109 such events in the regional catalogue. For 102 earthquakes the SMT and depth of the equivalent point source were calculated. The calculations were performed for the SMT double-couple model using a nonlinear algorithm. The level of seismicity according to the “SOUS'09” scale in 2016 corresponds to the “background increased”, for 2017 it is also the “background increased”, but within the assessment accuracy – “high”. In 2016–2017 within the Kamchatka branch area of responsibility, an atypical pattern of the location of earthquake epicenters was observed due to the occurrence of two strong events – the Near Aleutian (Mw=7.8, July 17, 2017) and South Ozernovsky (Mw=6.6, March 29, 2017) earthquakes and their aftershock processes.

We classify 2 + 1 dimensional integrable systems with nonlocality of the intermediate long wave type. Links to the 2 + 1 dimensional waterbag system are established. Dimensional reductions of integrable systems constructed in this paper provide dispersive regularisations of hydrodynamic equations governing propagation of long nonlinear waves in a shear flow with piecewise linear velocity profile (for special values of vorticities).

Two classes of multi-phase algebro-geometric solutions of Mikhalëv equation are constructed. The first class of solutions is associated with the Korteweg-de-Vries (KdV) equation. The second one is related to the solutions of the Kaup-Boussinesq (KB) equation. We have established interrelations among the multi-soliton, trigonometric, rational, elliptic and other known solutions of the KdV and KB equations and the solutions of Mikhalëv equations. We show that the number of linearly independent finite-gap solutions of Mikhalëv system is equal to the number of phases of these solutions. For each class of solutions we have constructed examples of explicit solutions of Mikhalëv equation. In the previous works cited below the solutions of the Mikhalëv system were described implicitly, being reduced to the solutions of appropriate Jacobi inversion problems. Here, to solve the Mikhalëv system explicitly, we used the formalism of Baker-Akhiezer functions.

The aim of this article is to classify pairs of the first-order Hamiltonian operators of Dubrovin–Novikov type such that one of them has a non-local part defined by an isometry of its leading coefficient. An example of such a bi-Hamiltonian pair was recently found for the constant astigmatism equation. We obtain a classification in the case of two dependent variables, and a significant new example with three dependent variables that is an extension of a hydrodynamic-type system obtained from a particular solution of the Witten–Dijkgraaf–Verlinde–Verlinde equations.

Recently a classification of contactly-nonequivalent three-dimensional linearly degenerate equations of the second order was presented by E.V. Ferapontov and J. Moss.The equations are Lax-integrable.In our paper we prove that all these equations are connected with each other by appropriate Bäcklund transformations.

Abstract—The digital seismic network in Kamchatka deployed in 2006–2010 provided a fundamental possibility for calculating seismic moment tensor (SMT) of Kamchatka earthquakes from broadband waveforms recorded by regional stations. The paper describes the method for calculating SMT for the earthquakes with pure double couple (DC) mechanism based on the DC tensor decomposition and explicit partial linear inversion in two (of four) variables. This allows the least square objective function to be expressed in terms of two angles specifying the orientation of the neutral (zero) axis of the DC tensor. The inversion with respect to the angles is conducted using the Levenberg–Marquardt iterative method. This technique was used for calculating focal mechanisms, moment magnitudes, and equivalent point-source depths for 31 Kamchatka earthquakes with moment magnitudes Mw = 4.3–6.2. The comparison of the obtained estimates with the corresponding GCMT catalog data which exist for 19 events has shown fairly reasonable agreement between the results. The angle K characterizing the discrepancy between the DC mechanism calculated directly from the waveforms in our study and the best DC mechanism constructed from GCMT null-trace tensor is at most 25° for 16 events of 19. The moment magnitudes are underestimated by ~0.1 relative to the GCMT counterparts and the depths are underestimated by 8 km on average.

In this paper, we consider a new class of Hamiltonian hydrodynamic type systems whose conservation laws are polynomial with respect to one of the field variables.

—The Udina volcanic complex located in the southeastern part of the Klyuchevskoy group of volcanoes in Kamchatka remained dormant for several thousand years, but the magmatic system beneath the area may be awakening judging by seismic unrest. Seismicity in the area is characterized by data from permanent regional seismic stations and campaign local stations, as well as by data of the Kamchatka earthquake catalog. Seismic activity having nucleated at shallow depths in the vicinities of the Udina volcanoes since October 2017 may reflect a beginning cycle of volcanism. The earthquakes are mainly long-period (LP) 0.5–5 Hz events, which are commonly attributed to the movement of viscous magma and resonance phenomena in magma conduits. Such earthquakes may be a response to inputs of new magma batches to the plumbing system that feeds the volcanoes and thus may be precursors of volcanic unrest. Seismic campaigns of May–July 2018 near the Udina complex provided more rigorous constraints on earthquake coordinates and origin depths and showed that most of the earthquakes originated within 5 km beneath the Bolshaya Udina Volcano. Seismic tomographic inversion using the LOTOS code revealed a zone of high P-wave velocities, low S-wave velocities, and a high vP/vS ratio directly beneath the volcano. Such a combination of parameters typically occurs in active volcanic areas and marks intrusion of partially molten magma and/or liquid fluids. The velocity anomaly detected in 2018 is shallower than that recovered in 2014–2015. The seismic evidence, along with the available geological and geophysical data, record the movement of viscous magma related to the Udina feeding system in the middle crust, which is implicit proof for connection between the intermediate crustal and deep mantle magma sources renewed after a long lull.

We investigate the integrability of Euler–Lagrange equations associated with 2D second-order Lagrangians of the form $$\begin{aligned} \int f(u_{xx},u_{xy},u_{yy})\ \mathrm{d}x\mathrm{d}y. \end{aligned}$$ By deriving integrability conditions for the Lagrangian density f, examples of integrable Lagrangians expressible via elementary functions, Jacobi theta functions and dilogarithms are constructed. A link of second-order integrable Lagrangians to WDVV equations is established. Generalisations to 3D second-order integrable Lagrangians are also discussed.

Показано, что система нелинейных уравнений двухфотонного распространения света с действительными амплитудами огибающих может быть решена в общем виде классическим методом Лиувилля. Эта система, как и другие аналогичные системы интегрируемых по Дарбу уравнений, связана с модифицированным уравнением Лиувилля, и найденное решение дает также общие решения таких модифицированных уравнений. Сделан вывод, что метод Лиувилля предоставляет эффективный способ интегрирования класса конкретных систем, допускающих интегрирование по Дарбу.