arXiv subjects

Z. A. Sobirov

Publications and source records attributed to Z. A. Sobirov.

8 recordsLinked to original sources

arXiv2026

A generalized time-fractional Kuramoto-Sivashinsky equation in the Schwartz space: global solvability and stability

We study the Cauchy problem for a generalized time-fractional Kuramoto--Sivashinsky equation on $\mathbb R$ with the regularized Caputo derivative in the Schwartz space. The linear problem is solved by the Fourier transform and a Mittag--Leffler representation, with preservation of the Schwartz class. For the nonlinear problem, local solvability is established by successive approximations with convergence in the full Schwartz topology. A continuation principle accounting for the complete fractional memory is developed: the history term is estimated uniformly with respect to the continuation point, which yields global solvability on every finite time interval. An $L^2(\mathbb R)$ stability estimate is also obtained, implying uniqueness.

math.AP↗

arXiv2025

Fractional Sturm-Liouville problem on metric graphs

In the present paper, we investigate the fractional analog of the Sturm-Liouville problem on a metric graph using a combination of left Riemann-Liouville and right Caputo fractional derivatives. This combination creates a symmetric and positive analog of the Sturm-Liouville operator. We demonstrated that the operator has a countable number of eigenvalues converging to infinity and analyzed the convergence of the series of the reciprocal eigenvalues, providing estimates for the eigenfunctions.

math.AP↗

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↗

arXiv2011

Stationary Nonlinear Schrödinger Equation on Simplest Graphs: Boundary conditions and exact solutions

We treat the stationary (cubic) nonlinear Schrödinger equation (NSLE) on simplest graphs. Formulation of the problem and exact analytical solutions of NLSE are presented for star graphs consisting of three bonds. It is shown that the method can be extended for the case of arbitrary number of bonds of star graphs and for other simplest topologies such as tree and loop graphs. The case of repulsive and attractive nonlinearities are treated separately.

nlin.SI↗

arXiv2011

Time dependent neutrino billiards

Quantum dynamica of a massless Dirac particle in time-dependent 1D box and circular billiard with time-dependent radius is studied. An exact analytical wave functions and eigenvalues are obtained for the case of linear time-dependence of the boundary position.

quant-ph↗

arXiv2011

Cauchy Problem for for some high order generalization of Korteweg - de Vries equation

In this work we study Cauchy problem for a high-order differential equation $\frac{\partial u(y,x)}{\partial y}+P(\frac{\partial}{\partial x})u(y,x)=γ\frac{\partial}{\partial x}(u^2(y,x))+F(y,x)$. We prove that the problem is well-posed both for linear ($γ=0$) and nonlinear equations on the class of rapidly decaying Schwartz functions. Furthermore, for the case when the initial condition is given on $L_2(\mathbf{R}^1)$ we prove the existence of the unique solution on the space $L_{\infty}(0,y_0; L_2(\mathbf{R}^1))\bigcap L_2(0,y_0; H^{n-1}(\mathbf{R}^1))\bigcap L_2(0,y_0;H^{n}(-r, r))$, where $r$ is an arbitrary positive number. It is also shown that the solution continuously depends on the initial conditions.

math-ph↗

arXiv2011

Ideal quantum gas in expanding cavity: nature of non-adiabatic force

We consider a quantum gas of non-interacting particles confined in the expanding cavity, and investigate the nature of the non-adiabatic force which is generated from the gas and acts on the cavity wall. Firstly, with use of the time-dependent canonical transformation which transforms the expanding cavity to the non-expanding one, we can define the force operator. Secondly, applying the perturbative theory which works when the cavity wall begins to move at time origin, we find that the non-adiabatic force is quadratic in the wall velocity and thereby does not break the time-reversal symmetry, in contrast with the general belief. Finally, using an assembly of the transitionless quantum states, we obtain the nonadiabatic force exactly. The exact result justifies the validity of both the definition of force operator and the issue of the perturbative theory. The mysterious mechanism of nonadiabatic transition with use of transitionless quantum states is also explained. The study is done on both cases of the hard-wall and soft-wall confinement with the time-dependent confining length.

quant-ph↗

arXiv2011

Transport in simple networks described by integrable discrete nonlinear SchrÄodinger equation

We elucidate the case in which the Ablowitz-Ladik (AL) type discrete nonlinear SchrÄodinger equa- tion (NLSE) on simple networks (e.g., star graphs and tree graphs) becomes completely integrable just as in the case of a simple 1-dimensional (1-d) discrete chain. The strength of cubic nonlinearity is different from bond to bond, and networks are assumed to have at least two semi-infinite bonds with one of them working as an incoming bond. The present work is a nontrivial extension of our preceding one (Sobirov et al, Phys. Rev. E 81, 066602 (2010)) on the continuum NLSE to the discrete case. We find: (1) the solution on each bond is a part of the universal (bond-independent) AL soliton solution on the 1-d discrete chain, but is multiplied by the inverse of square root of bond-dependent nonlinearity; (2) nonlinearities at individual bonds around each vertex must satisfy a sum rule; (3) under findings (1) and (2), there exist an infinite number of constants of motion. As a practical issue, with use of AL soliton injected through the incoming bond, we obtain transmission probabilities inversely proportional to the strength of nonlinearity on the outgoing bonds.

math-ph↗