Hodograph Method and Numerical Integration of Two Hyperbolic Quasilinear Equations. Part I. The Shallow Water Equations

In paper [S.I. Senashov, A. Yakhno. 2012. SIGMA. Vol.8. 071] the variant of the hodograph method based on the conservation laws for two hyperbolic quasilinear equations of the first order is described. Using these results we propose a method which allows to reduce the Cauchy problem for the two quasilinear PDE's to the Cauchy problem for ODE's. The proposed method is actually some similar method of characteristics for a system of two hyperbolic quasilinear equations. The method can be used effectively in all cases, when the linear hyperbolic equation in partial derivatives of the second order with variable coefficients, resulting from the application of the hodograph method, has an explicit expression for the Riemann-Green function. One of the method's features is the possibility to construct a multi-valued solutions. In this paper we present examples of method application for solving the classical shallow water equations.Comment: 19 pages, 5 figure

Hodograph Method and Numerical Solution of the Two Hyperbolic Quasilinear Equations. Part III. Two-Beam Reduction of the Dense Soliton Gas Equations

The paper presents the solutions for the two-beam reduction of the dense soliton gas equations (or Born-Infeld equation) obtained by analytical and numerical methods. The method proposed by the authors is used. This method allows to reduce the Cauchy problem for two hyperbolic quasilinear PDEs to the Cauchy problem for ODEs. In some respect, this method is analogous to the method of characteristics for two hyperbolic equations. The method is effectively applicable in all cases when the explicit expression for the Riemann-Green function for some linear second order PDE, resulting from the use of the hodograph method for the original equations, is known. The numerical results for the two-beam reduction of the dense soliton gas equations, and the shallow water equations (omitting in the previous papers) are presented. For computing we use the different initial data (periodic, wave packet).Comment: 22 pages, 11 figures. arXiv admin note: substantial text overlap with arXiv:1503.0176

Mathematical Model of a pH-gradient Creation at Isoelectrofocusing. Part II. Numerical Solution of the Stationary Problem

The mathematical model describing the natural textrm{pH}-gradient arising under the action of an electric field in an aqueous solution of ampholytes (amino acids) is constructed and investigated. This paper is the second part of the series papers \cite{Part1,Part3,Part4} that are devoted to pH-gradient creation problem. We present the numerical solution of the stationary problem. The equations system has a small parameter at higher derivatives and the turning points, so called stiff problem. To solve this problem numerically we use the shooting method: transformation of the boundary value problem to the Cauchy problem. At large voltage or electric current density we compare the numerical solution with weak solution presented in Part 1.Comment: 14 pages, 8 figure

Modeling of zonal electrophoresis in plane channel of complex shape

The zonal electrophoresis in the channels of complex forms is considered mathematically with the use of computations. We show that for plane S-type rectangular channels stagnation regions can appear that cause the strong variations of the spatial distribution of an admixture. Besides, the shape of an admixture zone is strongly influenced by the effects of electromigration and by a convective mixing. Taking into account the zone spreading caused by electromigration, the influence of vertex points of cannel walls, and convection would explain the results of electrophoretic experiments, which are difficult to understand otherwise.Comment: 13 pages, 10 figure

Rotating electrohydrodynamic flow in a suspended liquid film

The mathematical model of a rotating electrohydrodynamic flow in a thin suspended liquid film is proposed and studied. The motion is driven by the given difference of potentials in one direction and constant external electrical field \vE_\text{out} in another direction in the plane of a film. To derive the model we employ the spatial averaging over the normal coordinate to a film that leads to the average Reynolds stress that is proportional to |\vE_\text{out}|^3. This stress generates tangential velocity in the vicinity of the edges of a film that, in turn, causes the rotational motion of a liquid. The proposed model is aimed to explain the experimental observations of the \emph{liquid film motor} (see arXiv:0805.0490v2).Comment: 12 pages, 9 figures. (Submitted to Phys. Rev. E

Mathematical Model of a pH-gradient Creation at Isoelectrofocusing

The mathematical model describing the stationary natural pH-gradient arising under the action of an electric field in an aqueous solution of ampholytes (amino acids) is constructed and investigated. The model is a part of a more general model of the isoelectrofocusing process. Investigation is based on the approximation of a weak solution by the piecewise continuous non-smooth functions. The method can be used for solving classes of problems for ODEs with a small parameter at higher derivatives and the turning points.Comment: 29 pages, 6 figure

Mathematical Model of a pH-gradient Creation at Isoelectrofocusing. Part IV. Theory

The mathematical model describing the non-stationary natural pH-gradient arising under the action of an electric field in an aqueous solution of ampholytes (amino acids) is constructed. The model is a part of a more general model of the isoelectrofocusing (IEF) process. The presented model takes into account: 1) general Ohm's law (electric current flux includes the diffusive electric current); 2) dissociation of water; 3) difference between isoelectric point (IEP) and isoionic point (PZC -- point of zero charge). We also study the Kohlraush's function evolution and discuss the role of the Poisson-Boltzmann equation.Comment: 15 pages, 1 figur

Interactions between discontinuities for binary mixture separation problem and hodograph method

The Cauchy problem for first-order PDE with the initial data which have a piecewise discontinuities localized in different spatial points is completely solved. The interactions between discontinuities arising after breakup of initial discontinuities are studied with the help of the hodograph method. The solution is constructed in analytical implicit form. To recovery the explicit form of solution we propose the transformation of the PDEs into some ODEs on the level lines (isochrones) of implicit solution. In particular, this method allows us to solve the Goursat problem with initial data on characteristics. The paper describes a specific problem for zone electrophoresis (method of the mixture separation). However, the method proposed allows to solve any system of two first-order quasilinear PDEs for which the second order linear PDE, arising after the hodograph transformation, has the Riemann-Green function in explicit form.Comment: 19 pages, 11 figure

Modeling dynamics of entangled physical systems with superconducting quantum computer

We implement several quantum algorithms in real five-qubit superconducting quantum processor IBMqx4 to perform quantum computation of the dynamics of spin-1/2 particles interacting directly and indirectly through the boson field. Particularly, we focus on effects arising due to the presence of entanglement in the initial state of the system. The dynamics is implemented in a digital way using Trotter expansion of evolution operator. Our results demonstrate that dynamics in our modeling based on real device is governed by quantum interference effects being highly sensitive to phase parameters of the initial state. We also discuss limitations of our approach due to the device imperfection as well as possible scaling towards larger systems.Comment: 11 page

Algorithmic simulation of far-from-equilibrium dynamics using quantum computer

We point out that superconducting quantum computers are prospective for the simulation of the dynamics of spin models far from equilibrium, including nonadiabatic phenomena and quenches. The important advantage of these machines is that they are programmable, so that different spin models can be simulated in the same chip, as well as various initial states can be encoded into it in a controllable way. This opens an opportunity to use superconducting quantum computers in studies of fundamental problems of statistical physics such as the absence or presence of thermalization in the free evolution of a closed quantum system depending on the choice of the initial state as well as on the integrability of the model. In the present paper, we performed proof-of-principle digital simulations of two spin models, which are the central spin model and the transverse-field Ising model, using 5- and 16-qubit superconducting quantum computers of the IBM Quantum Experience. We found that these devices are able to reproduce some important consequences of the symmetry of the initial state for the system's subsequent dynamics, such as the excitation blockade. However, lengths of algorithms are currently limited due to quantum gate errors. We also discuss some heuristic methods which can be used to extract valuable information from the imperfect experimental data.Comment: 26 pages. arXiv admin note: substantial text overlap with arXiv:1710.0965