arXiv subjects

C. B. Muratov

Publications and source records attributed to C. B. Muratov.

At least 19 recordsLinked to original sources

arXiv2025

Skyrmion stacking in stray field-coupled ultrathin ferromagnetic multilayers

This paper explores the energy landscape of ferromagnetic multilayer heterostructures that feature magnetic skyrmions -- tiny whirls of spins with non-trivial topology -- in each magnetic layer. Such magnetic heterostructures have been recently pursued as possible hosts of room temperature stable magnetic skyrmions suitable for the next generation of low power information technologies and unconventional computing. The presence of stacked skyrmions in the adjacent layers gives rise to a strongly coupled nonlinear system, whereby the induced magnetic field plays a crucial stabilizing role. Starting with the micromagnetic modeling framework, we derive a general reduced energy functional for a fixed number of ultrathin ferromagnetic layers with perpendicular magnetocrystalline anisotropy. We next investigate this energy functional in the regime in which the energy is dominated by the intralayer exchange interaction and formally obtain a finite-dimensional description governed by the energy of a system of one skyrmion per layer as a function of the position, radius and the rotation angle of each of theses skyrmions. For the latter, we prove that energy minimizers exist for all fixed skyrmion locations. We then focus on the simplest case of stray field-coupled ferromagnetic bilayers and completely characterize the energy minimizers. We show that the global energy minimizers exist and consist of two stray field-stabilized Néel skyrmions with antiparallel in-plane magnetization components. We also calculate the energy of two skyrmions of equal radius as a function of their separation distance.

math.AP↗

arXiv2024

Reduced energies for thin ferromagnetic films with perpendicular anisotropy

We derive four reduced two-dimensional models that describe, at different spatial scales, the micromagnetics of ultrathin ferromagnetic materials of finite spatial extent featuring perpendicular magnetic anisotropy and interfacial Dzyaloshinskii-Moriya interaction. Starting with a microscopic model that regularizes the stray field near the material's lateral edges, we carry out an asymptotic analysis of the energy by means of $Γ$-convergence. Depending on the scaling assumptions on the size of the material domain vs. the strength of dipolar interaction, we obtain a hierarchy of the limit energies that exhibit progressively stronger stray field effects of the material edges. These limit energies feature, respectively, a renormalization of the out-of-plane anisotropy, an additional local boundary penalty term forcing out-of-plane alignment of the magnetization at the edge, a pinned magnetization at the edge, and, finally, a pinned magnetization and an additional field-like term that blows up at the edge, as the sample's lateral size is increased. The pinning of the magnetization at the edge restores the topological protection and enables the existence of magnetic skyrmions in bounded samples.

math.AP↗

arXiv2021

Transverse domain walls in thin ferromagnetic strips

We present a characterization of the domain wall solutions arising as minimizers of an energy functional obtained in a suitable asymptotic regime of micromagnetics for infinitely long thin film ferromagnetic strips in which the magnetization is forced to lie in the film plane. For the considered energy, we provide existence, uniqueness, monotonicity, and symmetry of the magnetization profiles in the form of 180$^\circ$ and 360$^\circ$ walls. We also demonstrate how this energy arises as a $Γ$-limit of the reduced two-dimensional thin film micromagnetic energy that captures the non-local effects associated with the stray field, and characterize its respective energy minimizers.

math.AP↗

arXiv2016

Threshold phenomena for symmetric-decreasing radial solutions of reaction-diffusion equations

We study the long time behavior of positive solutions of the Cauchy problem for nonlinear reaction-diffusion equations in $\mathbb{R}^N$ with bistable, ignition or monostable nonlinearities that exhibit threshold behavior. For $L^2$ initial data that are radial and non-increasing as a function of the distance to the origin, we characterize the ignition behavior in terms of the long time behavior of the energy associated with the solution. We then use this characterization to establish existence of a sharp threshold for monotone families of initial data in the considered class under various assumptions on the nonlinearities and spatial dimension. We also prove that for more general initial data that are sufficiently localized the solutions that exhibit ignition behavior propagate in all directions with the asymptotic speed equal to that of the unique one-dimensional variational traveling wave.

math.AP↗

arXiv2014

Front propagation in geometric and phase field models of stratified media

We study front propagation problems for forced mean curvature flows and their phase field variants that take place in stratified media, i.e., heterogeneous media whose characteristics do not vary in one direction. We consider phase change fronts in infinite cylinders whose axis coincides with the symmetry axis of the medium. Using the recently developed variational approaches, we provide a convergence result relating asymptotic in time front propagation in the diffuse interface case to that in the sharp interface case, for suitably balanced nonlinearities of Allen-Cahn type. The result is established by using arguments in the spirit of $Γ$-convergence, to obtain a correspondence between the minimizers of an exponentially weighted Ginzburg-Landau type functional and the minimizers of an exponentially weighted area type functional. These minimizers yield the fastest traveling waves invading a given stable equilibrium in the respective models and determine the asymptotic propagation speeds for front-like initial data. We further show that generically these fronts are the exponentially stable global attractors for this kind of initial data and give sufficient conditions under which complete phase change occurs via the formation of the considered fronts.

math.AP↗

arXiv2012

Threshold phenomena for symmetric decreasing solutions of reaction-diffusion equations

We study the long time behavior of solutions of the Cauchy problem for nonlinear reaction-diffusion equations in one space dimension with the nonlinearity of bistable, ignition or monostable type. We prove a one-to-one relation between the long time behavior of the solution and the limit value of its energy for symmetric decreasing initial data in $L^2$ under minimal assumptions on the nonlinearities. The obtained relation allows to establish sharp threshold results between propagation and extinction for monotone families of initial data in the considered general setting.

math.AP↗

arXiv2011

Multiplicity of supercritical fronts for reaction-diffusion equations in cylinders

We study multiplicity of the supercritical traveling front solutions for scalar reaction-diffusion equations in infinite cylinders which invade a linearly unstable equilibrium. These equations are known to possess traveling wave solutions connecting an unstable equilibrium to the closest stable equilibrium for all speeds exceeding a critical value. We show that these are, in fact, the only traveling front solutions in the considered problems for sufficiently large speeds. In addition, we show that other traveling fronts connecting to the unstable equilibrium may exist in a certain range of the wave speed. These results are obtained with the help of a variational characterization of such solutions.

math.AP↗

arXiv2011

Global exponential convergence to variational traveling waves in cylinders

We prove, under generic assumptions, that the special variational traveling wave that minimizes the exponentially weighted Ginzburg-Landau functional associated with scalar reaction-diffusion equations in infinite cylinders is the long-time attractor for the solutions of the initial value problems with front-like initial data. The convergence to this traveling wave is exponentially fast. The obtained result is mainly a consequence of the gradient flow structure of the considered equation in the exponentially weighted spaces and does not depend on the precise details of the problem. It strengthens our earlier generic propagation and selection result for "pushed" fronts.

math.AP↗

arXiv2011

Cavitation-induced ignition of cryogenic hydrogen-oxygen fluids

The Challenger disaster and purposeful experiments with liquid hydrogen (H2) and oxygen (Ox) tanks demonstrated that cryogenic H2/Ox fluids always self-ignite in the process of their mixing. Here we propose a cavitation-induced self-ignition mechanism that may be realized under these conditions. In one possible scenario, self-ignition is caused by the strong shock waves generated by the collapse of pure Ox vapor bubble near the surface of the Ox liquid that may initiate detonation of the gaseous H2/Ox mixture adjacent to the gas-liquid interface. This effect is further enhanced by H2/Ox combustion inside the collapsing bubble in the presence of admixed H2 gas.

physics.flu-dyn↗

arXiv2010

Domain structure of bulk ferromagnetic crystals in applied fields near saturation

We investigate the ground state of a uniaxial ferromagnetic plate with perpendicular easy axis and subject to an applied magnetic field normal to the plate. Our interest is the asymptotic behavior of the energy in macroscopically large samples near the saturation field. We establish the scaling of the critical value of the applied field strength below saturation at which the ground state changes from the uniform to a branched domain magnetization pattern and the leading order scaling behavior of the minimal energy. Furthermore, we derive a reduced sharp-interface energy giving the precise asymptotic behavior of the minimal energy in macroscopically large plates under a physically reasonable assumption of small deviations of the magnetization from the easy axis away from domain walls. On the basis of the reduced energy, and by a formal asymptotic analysis near the transition, we derive the precise asymptotic values of the critical field strength at which non-trivial minimizers (either local or global) emerge. The non-trivial minimal energy scaling is achieved by magnetization patterns consisting of long slender needle-like domains of magnetization opposing the applied field

nlin.PS↗

arXiv2008

Geometric strong segregation theory for compositionally asymmetric diblock copolymer melts

We have identified the effect of the Wigner-Seitz cell geometry in the strong segregation limit of diblock copolymer melts with strong composition asymmetry. A variational problem is proposed describing the distortions of the chain paths due to the geometric constraints imposed by the cell shape. We computed the geometric excess energies for cylindrical phases arranged into hexagonal, square, and triangular lattices and explicitly demonstrated that the hexagonal lattice has the lowest energy for a fixed cell area.

cond-mat.soft↗

arXiv2008

Bit storage by $360^\circ$ domain walls in ferromagnetic nanorings

We propose a design for the magnetic memory cell which allows an efficient storage, recording, and readout of information on the basis of thin film ferromagnetic nanorings. The information bit is represented by the polarity of a stable 360$^\circ$ domain wall introduced into the ring. Switching between the two magnetization states is achieved by the current applied to a wire passing through the ring, whereby the $360^\circ$ domain wall splits into two charged $180^\circ$ walls, which then move to the opposite extreme of the ring to recombine into a $360^\circ$ wall of the opposite polarity.

cond-mat.mtrl-sci↗

arXiv2002

A quantitative approximation scheme for the traveling wave solutions in the Hodgkin-Huxley model

We introduce an approximation scheme for the Hodgkin-Huxley model of nerve conductance which allows to calculate both the speed of the traveling pulses and their shape in quantitative agreement with the solutions of the model. We demonstrate that the reduced problem for the front of the traveling pulse admits a unique solution. We obtain an explicit analytical expression for the speed of the pulses which is valid with good accuracy in a wide range of the parameters.

nlin.PS↗

arXiv2002

Theory of domain patterns in systems with long-range interactions of Coulomb type

We develop a theory of the domain patterns in systems with competing short-range attractive interactions and long range repulsive Coulomb interactions. We take an energetic approach, in which patterns are considered as critical points of a mean-field free energy functional. Close to the microphase separation transition, this functional takes on a universal form, allowing to treat a number of diverse physical situations within a unified framework. We use asymptotic analysis to study domain patterns with sharp interfaces. We derived an interfacial representation of the pattern's free energy which remains valid in the fluctuating system, with a suitable renormalization of the Coulomb interaction's coupling constant. We also derived integrodifferential equations describing the stationary domain patterns of arbitrary shapes and their thermodynamic stability, coming from the first and second variation of the interfacial free energy. We showed that the length scale of a stable domain pattern must obey a certain scaling law with the strength of the Coulomb interaction. We analyzed existence and stability of localized (spots, stripes, annuli) and periodic (lamellar, hexagonal) patterns in two dimensions. We showed that these patterns are metastable in certain ranges of the parameters and that they can undergo morphological instabilities leading to the formation of more complex patterns. We discuss nucleation of the domain patterns by thermal fluctuations and pattern formation scenarios for various thermal quenches. We argue that self-induced disorder is an intrinsic property of the domain patterns in the systems under consideration.

cond-mat.soft↗

arXiv2000

Spike autosolitons in the Gray-Scott model

We performed a comprehensive study of the spike autosolitons: self-sustained solitary inhomogeneous states, in the classical reaction-diffusion system --- the Gray-Scott model. We developed singular perturbation techniques based on the strong separation of the length scales to construct asymptotically the solutions in the form of a one-dimensional static autosolitons, higher-dimensional radially-symmetric static autosolitons, and two types of traveling autosolitons. We studied the stability of the static autosolitons in one and three dimensions and analyzed the properties of the static and the traveling autosolitons.

patt-sol↗

arXiv1999

Traveling wave solutions in the Burridge-Knopoff model

The slider-block Burridge-Knopoff model with the Coulomb friction law is studied as an excitable medium. It is shown that in the continuum limit the system admits solutions in the form of the self-sustained shock waves traveling with constant speed which depends only on the amount of the accumulated stress in front of the wave. For a wide class of initial conditions the behavior of the system is determined by these shock waves and the dynamics of the system can be expressed in terms of their motion. The solutions in the form of the periodic wave trains and sources of counter-propagating waves are analyzed. It is argued that depending on the initial conditions the system will either tend to synchronize or exhibit chaotic spatiotemporal behavior.

patt-sol↗

arXiv1998

Unusual Coarsening During Phase Separation in Polymer Systems

We introduce a kinetic model of coarsening of transient polymer networks during the intermediate stages of phase separation in polymer systems. The model explicitly takes into account the effect of the connectivity of polymer chains. We show that during the intermediate stages the size of the droplet grows according to an unusual coarsening law $R \sim t^{1/(d + 3)}$. When the network structure breaks up, the coarsening law may cross over to $R \sim \ln t^{-1}$, until only at very late stages the Lifshitz-Slyozov coarsening law $R \sim t^{1/3}$ is recovered.

cond-mat.mtrl-sci↗

arXiv1998

Theory of Spike Spiral Waves in a Reaction-Diffusion System

We discovered a new type of spiral wave solutions in reaction-diffusion systems --- spike spiral wave, which significantly differs from spiral waves observed in FitzHugh-Nagumo-type models. We present an asymptotic theory of these waves in Gray-Scott model. We derive the kinematic relations describing the shape of this spiral and find the dependence of its main parameters on the control parameters. The theory does not rely on the specific features of Gray-Scott model and thus is expected to be applicable to a broad range of reaction-diffusion systems.

patt-sol↗