#### Aim of the project

The aim of this work is to develop numerical and asymptotic methods for studying complex physical problems, including nonlinear ones. The work consists of 5 parts, each of which develops research methods in specific subject fields.

The aim of the first part of the work is to study new non-Markov effects in the quantum dynamics of reservoir-related systems (study of the effect of reservoir memory).

The aim of the second part of the work is to study classical propagation models, as well as the influence of the structural properties of complex networks on the characteristics of dynamic processes: characteristic time, the maximum number of infected ones, and the development of models that simulate higher-order interactions in propagation processes (simplicial interaction).

The aim of the third part of the work is to study the problems of flow of a viscous incompressible liquid around a semi-infinite plate with small localized or periodic irregularities on its surface whose shape depends on time.

The aim of the fourth part of the work is to develop general methods for constructing effective Hamiltonians describing the motion of quantum particles in problems with symmetries and resonance in the leading part of the Hamiltonian and to construct asymptotic spectral series and asymptotic eigenfunctions for the corresponding quantum problems.

The aim of the fifth part of the paper is to develop weak asymptotics methods by studying, as an example, the problem of interaction of delta-shock waves in the system of equations of gas dynamics without pressure (continuity equations and momentum conservation equations).

#### Methods used in the project

In the first part of the work, we use the Popov-Fedotov semionic formulation, which reduces the spin dynamics to the dynamics of a two-time complex quantum field after the Hubbard-Stratonovich transformation. For this field, a dissipative action is obtained and, after its expansion in a perturbation series with respect to the interaction with the reservoir, a diagram technique is developed that allows summing the perturbation series. As a special case, a spin diagram is obtained. To analyze the dynamics of formation of the Bose condensate of exciton polaritons, the first-principle stochastic integro-differential equations describing it are used. The Langevin-type equations with quantum noise, which are obtained from them, are solved numerically with a set of statistics for averaging over the noise realizations.

In the second part of the work, we use computer simulation methods including probabilistic Monte Carlo methods, various algorithms on graphs, algorithms for searching communities in graphs (Newman methods, spectral analysis), algorithms for storing and treating big data, and various network visualization algorithms. For theoretical estimates, a standard set of methods of statistical physics is used, including the generating function method, mean field approximation, cluster analysis, and random matrix theory.

In the third part of the work, the solutions are constructed using the method of multiscale asymptotic analysis based on a combination of boundary-layer decomposition methods and the stabilization (localization) method; and for numerical simulation of the flow, the methods of the theory of difference schemes are used.

In the fourth part of this work, to construct asymptotic spectral series, we use algebraic methods for operator averaging, methods of noncommutative analysis, and the calculus of the symbol algebra, as well as methods of quasi-classical approximation.

In the fifth part of the work, the weak asymptotics method is used, in particular, the apparatus of finitely generated algebras asymptotic in a weak sense

#### Empirical base of research

The results of research on the subject of the project that were published in the world scientific literature were used as the empirical base of research.

#### Obtained results

In the first part of this work, we developed the formalism of semionic and two-time quantum fields to describe the non-Markov dynamics of spins in a heat reservoir. Based on this formalism, a diagram technique was developed that allows calculating Green's functions (correlators) of spin. It was shown that the standard equations for the dynamics of a two-level system used in quantum optics can be obtained as a special case of the developed model. It is proved that the Bloch-Redfield classical relation for the spin correlator in a thermostat in the low-frequency limit is universal. The non-Markov dynamics of formation of the Bose condensate of exciton polaritons interacting with a reservoir of supercondensate excitons is also considered. It is shown that non-Markov effects due to the reservoir memory can radically change the nature of condensate dynamics. In some ranges of system parameters, nonlinear relaxation oscillations occur with the exchange of excitons between the reservoir and the condensate. At higher pumping power, a chaotic regime occurs with sporadic transitions of the system between the lower and upper polariton branches. It is also shown that non-Markov effects reduce the pumping power threshold after which the condensate is formed.’

In the second part of the work, it is demonstrated that the distribution of connections between communities plays a significant role in the spread of interactions (epidemics) in complex networks. It is shown that in clusterized networks, where a power-law distribution of intercluster connections naturally occurrs, the propagation is more slowly with a less pronounced peak (the classical SIR (susceptible/infected/recovered) model was considered). A new propagation model is constructed with two activation transfer mechanisms: along the edge (classical mechanism) and along the triangle. Each mechanism is described by its own activity transfer constant. We analyzed the dynamics on random Erdos-Renyi networks and showed the presence of the phase transition; namely, depending on the coefficient of activity transfer along the simplex in equilibrium, either all vertices are active or the activation is completely extinguished. The existence of hysteresis that is typical of the phase transitions of the first kind is numerically demonstrated. A theoretical description of the observed effects is proposed in the mean field approximation.

In the third part of the work, a formal asymptotic solution with the double-deck structure of boundary layer is constructed for problems of flow of a viscous incompressible liquid around a semi-infinite plate with small irregularities on its surface whose shape depends on time. The influence of two types of variation in the irregularity shape in the course of time (fluctuations in the amplitude and the motion by and against the flow) on the flow pattern, including the vortex formation process, is numerically studied.

In the fourth part of this work, we propose a new approach to calculating the coefficients in the quantum averaging procedure for the case of perturbed Hamiltonian of a multi-frequency resonant harmonic oscillator. Using the notion of twisted product introduced in this paper, the averaging procedure is transferred in the space of a graded symbol algebra. As a result, the averaged Hamiltonian is represented as a function of symmetries of the harmonic part of the Hamiltonian that form a quantum algebra. For the problem of motion of a quantum particle in a thin layer, it is shown that a small deformation of the layer leads to the splitting of degenerate energy levels and the appearance of geometric currents. The asymptotics of the spectrum and the eigenfunctions of the Hamiltonian of the system are constructed, and the dynamic equations for the coordinates of the leading center of charge are constructed. In the problem for a two-dimensional Hartree-type operator with a smooth self-action potential, asymptotic eigenvalues and asymptotic eigenfunctions are fetermined near the lower bounds of spectral clusters. Asymptotic formulas for quantum averages near the lower boundaries of spectral clusters are obtained.

In the fifth part of the work, a system of equations describing the interaction of delta-shock waves for equations of gas dynamics without pressure in divergence form in the one-dimensional case is studied numerically and analytically. A new type of interaction of solitary nonlinear waves, different from those known for solitons and kinks, has been discovered; namely, a pair of catching-up and overtaking delta-shock waves is formed in the interaction.

#### Degree of implementation, recommendations for implementation or results of implementation of the research results

The non-Markov character of the spin dynamics in the thermostat predicted in this work, which should take place on time scales of tens of picoseconds, can be verified in modern femtosecond spectroscopy experiments. The developed formalism of semions and two-time fields can be used to solve problems of analyzing the non-Markov dynamics of spin and qubit systems in a thermostat and is more General than the traditionally used Markov equations of quantum optics. Such tasks are relevant due to the development of quantum computer technologies and the appearance of possibilities of joint work with other controlled quantum systems. The apparatus of stochastic integro-differential equations describing the dynamics of formation of the Bose condensate of polaritons can be used to analyze the kinetics of Bose condensation in various systems and to study new nonlinear effects in condensate dynamics.

The presented models and results can shed light on the key properties of complex networks that determine the dynamics of the spread of activations and lead to new methods for identifying superspreaders (superspreaders in a complex network are the vertices that are most effective in transmitting information or epidemics) in the network. The latter task is closely related to the topological properties of individual vertices. Identification of superspreaders from structural properties of vertices is an open question. This task is important for both the effective fight against the spread of epidemics, and the tasks of identifying the most functionally important elements of biological systems: genes, proteins, and brain regions.

The obtained results of studying the flow around a plate with small irregularities whose shape depends on time can be used in various applied problems, for example, in problems of erosion of the surface in the flow, of chemical reaction between the flow and the surface, and the external influence on the elastic surface of the plate.

The obtained results can be applied in various tasks of construction and research of complex quantum mechanical systems, as well as of the development of quantum and nano-electronic devices.

The development of the theory of finitely generated asymptotic algebras can be used to describe various nonlinear interaction processes. In particular, a new mechanism of interaction discovered in this research can be found in the Zeldovich-Shandarin model used for macroscopic description of the mass distribution in the Universe. The results obtained can also be used, in particular, to construct weak approximations of generalized solutions to the Cauchy problem for hyperbolic conservation laws (A. Bressan’s front-tracking method).