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K. K. Sabirov

Publications and source records attributed to K. K. Sabirov.

At least 19 recordsLinked to original sources
arXiv2025

Kink-antikink soliton solutions of the nonlinear Klein-Gordon equation on branched structures

In this paper, we investigate the nonlinear Klein-Gordon equation on a metric star graph with three semi-infinite bonds. At the branching point, we impose a weighted continuity condition and a generalized weighted Kirchhoff condition for the derivatives of the wave function. By employing both analytical methods and numerical techniques, we construct exact and numerical soliton solutions that satisfy the vertex conditions and conserve energy and momentum. The results of analytic calculations are confirmed through numerical experiments, which demonstrate reflectionless propagation of kink-antikink soliton solutions. We compute and analyze the reflection coefficient, study the impact of various nonlinearity parameters, and further extend the formulation to other graph topologies, such as tree and loop graphs.

nlin.PS↗
arXiv2025

Burgers equation on networks: Metric graph based approach

Weconsider Burgers equation on metric graphs for simplest topologies such as star, loops, and tree graphs. Exact traveling wave solutions are obtained for the vertex boundary conditions providing mass conservation and continuity of the solution at the nodes. Constraints for the nonlinearity coefficients ensuring integrability of the Burgers equation are derived. Numerical treatment of the soliton dynamics and their transmission through the graph vertex is presented.

nlin.SI↗
arXiv2023

The nonlinear stationary differential equations with fractional order $1<α<2$ on metric star graphs

In this paper we consider nonlinear stationary fractional-in-space differential equations with order $1<α<2$ on the metric star graph with three finite bonds. At the branched point of the star graph we put the weight continuity and the generalized Kirchhoff rule. It is found the exact solutions of nonlinear stationary fractional equations on the star graph. These can be extended to the star graphs with any number of bonds.

math.CA↗
arXiv2022

Soliton generation in PT-symmetric optical fiber networks

We consider the problem of soliton generation in PT-symmetric optical fiber networks, where soliton dynamics is governed by nonlocal nonlinear Schrodinger equation on metric graphs. Exact formulae for the number of generated solitons are derived for the cases, when the problem is integrable. Numerical solutions are obtained for the case, when integrability is broken.

nlin.PS↗
arXiv2021

Dirac particles on periodic quantum graphs

We consider the Dirac equation on periodic networks (quantum graphs). The self-adjoint quasi periodic boundary conditions are derived. The secular equation allowing us to find the energy spectrum of the Dirac particles on periodic quantum graphs is obtained. Band spectra of the periodic quantum graphs of different topologies are calculated. Universality of the probability to be in the spectrum for certain graph topologies is observed.

quant-ph↗
arXiv2021

Reflectionless propagation of Manakov solitons on a line:A model based on the concept of transparent boundary conditions

We consider the problem of absence of backscattering in the transport of Manakov solitons on a line. The concept of transparent boundary conditions is used for modeling the reflectionless propagation of Manakov vector solitons in a one-dimensional domain. Artificial boundary conditions that ensure the absence of backscattering are derived and their numerical implementation is demonstrated.

nlin.SI↗
arXiv2020

Transparent boundary conditions for the sine-Gordon equation:Modeling the reflectionless propagation of kink solitons on a line

We consider the reflectionless transport of sine-Gordon solitons on a line. Transparent boundary conditions for the sine-Gordon equation on a line are derived using the so-called potential approach. Our numerical implementation of these novel boundary conditions proves the absence of the backscattering in transmission of sine-Gordon solitons through the boundary of the considered finite domains.

nlin.SI↗
arXiv2020

Soliton generation in optical fiber networks

We consider the problem of soliton generation in branched optical fibers. A model based on the nonlinear Schrodinger equation on metric graphs is proposed. Number of generated solitons is computed for different branching topologies considering different initial pulse profiles. Experimental realization of the model is discussed.

nlin.PS↗
arXiv2020

Dirac Particles in Transparent Quantum Graphs: Tunable transport of relativistic quasiparticles in branched structures

We consider the dynamics of relativistic spin-half particles in quantum graphs with transparent branching points. The system is modeled by combining the quantum graph concept with the one of transparent boundary conditions applied to the Dirac equation on metric graphs. Within such an approach, we derive simple constraints, which turn the usual Kirchhoff-type boundary conditions at the vertex equivalent to the transparent ones. Our method is applied to quantum star graph. An extension to more complicated graph topologies is straightforward.

quant-ph↗
arXiv2020

Exciton dynamics in branched conducting polymers: Quantum graphs based approach

We consider dynamics of excitons in branched conducting polymers. An effective model based on the use of quantum graph concept is applied for computing of exciton migration along the branched polymer chain Condition for the regime, when the transmission of exciton through the branching point is reflectionless is revealed.

cond-mat.mes-hall↗
arXiv2019

Transparent nonlinear networks

We consider the reflectionless transport of solitons in networks. The system is modeled in terms of the nonlinear Schrödinger equation on metric graphs, for which transparent boundary conditions at the branching points are imposed. This allows to derive simple constraints, which link equivalent usual Kirchhoff-type vertex conditions to the transparent ones. Our approach is applied to a metric star graph. An extension to more complicated graph topologies is straight forward.

nlin.PS↗
arXiv2019

Networks with point like nonlinearities

We study static nonlinear waves in networks described by a nonlinear Schrodinger equation with point-like nonlinearities on metric graphs. Explicit solutions fulfilling vertex boundary conditions are obtained. Spontaneous symmetry breaking caused by bifurcations is found.

nlin.PS↗
arXiv2018

Transparent Quantum Graphs

We consider quantum graphs with transparent branching points. To design such networks, the concept of transparent boundary conditions is applied to the derivation of the vertex boundary conditions for the linear Schrodinger equation on metric graphs. This allows to derive simple constraints, which use equivalent usual Kirchhoff-type boundary conditions at the vertex to the transparent ones. The approach is applied to quantum star and tree graphs. However, extension to more complicated graph topologies is rather straight forward.

quant-ph↗
arXiv2018

Dynamics of Dirac solitons in networks

We study dynamics of Dirac solitons in prototypical networks modeling them by the nonlinear Dirac equation on metric graphs. Soliton solutions of the nonlinear Dirac equation on simple metric graphs are obtained. It is shown that these solutions provide reflectionless vertex transmission of the Dirac solitons under suitable conditions. The constraints for bond nonlinearity coefficients, allowing reflectionless transmission over a Y-junction are derived. The analytical results are confirmed by direct numerical simulations.

nlin.PS↗
arXiv2018

PT-symmetric quantum graphs

We consider branched quantum wires, whose connection rules provide PT-symmetry for the Schrodinger equation on graph. For such PT-symmetric quantum graph we derive general boundary conditions which keep the Hamiltonian as PT-symmetric with real eigenvalues and positively defined norm. Explicit boundary conditions which are consistent with the general PT-symmetric boundary conditions are presented. Secular equations for finding the eigenvalues of the quantum graph are derived. Breaking of the Kirchhoff rule at the branching point is shown. Experimental realization of PT-symmetric quantum graphs on branched optical waveguides is discussed.

quant-ph↗
arXiv2018

Bogoliubov de Gennes equation on metric graphs

We consider Bogoliubov de Gennes equation on metric graphs. The vertex boundary conditions providing self-adjoint realization of the Bogoliubov de Gennes operator on a metric star graph are derived. Secular equation providing quantization of the energy and the vertex transmission matrix are also obtained. Application of the model for Majorana wire networks is discussed.

quant-ph↗
arXiv2012

Time-dependent quantum graph

In this paper we study quantum star graphs with time-dependent bond lengths. Quantum dynamics is treated by solving Schrodinger equation with time-dependent boundary conditions given on graphs. Time-dependence of the average kinetic energy is analyzed. Space-time evolution of the Gaussian wave packet is treated for harmonically breathing star graph.

cond-mat.mes-hall↗