Quantum control based on three forms of Lyapunov functions

This paper introduces the quantum control of Lyapunov functions based on the state distance, the mean of imaginary quantities and state errors.In this paper, the specific control laws under the three forms are given.Stability is analyzed by the La Salle invariance principle and the numerical simulation is carried out in a 2D test system.The calculation process for the Lyapunov function is based on a combination of the average of virtual mechanical quantities, the particle swarm algorithm and a simulated annealing algorithm.Finally, a unified form of the control laws under the three forms is given.

Oscillation properties of eigenfunctions for Sturm-Liouville problems with interface conditions via Prufer transformation

A class of Sturm-Liouville problems with discontinuity is studied in this paper.The oscillation properties of eigenfunctions for Sturm-Liouville problems with interface conditions are obtained.The main method used in this paper is based on Prufer transformation,which is different from the classical ones.Moreover,we give two examples to verify our main results.

RIEMANN-HILBERT PROBLEMS OF A SIX-COMPONENT MKDV SYSTEM AND ITS SOLITON SOLUTIONS

Based on a 4 x 4 matrix spectral problem, an AKNS soliton hierarchy with six potentials is generated. Associated with this spectral problem, a kind of Riemann-Hilbert problems is formulated for a six-component system of mKdV equations in the resulting AKNS hierarchy. Soliton solutions to the considered system of coupled mKdV equations are computed, through a reduced Riemann-Hilbert problem where an identity jump matrix is taken.

A numerical simulation of the generation and evolution of nonlinear internal waves across the Kara Strait

Nonlinear internal waves(NIWs) are ubiquitous around the Kara Sea, a part of the Arctic Ocean that is north of Siberia. Three hot spot sources for internal waves, one of which is the Kara Strait, have been identified based on Envisat ASAR. The generation and evolution of the NIWs through the interactions of the tide and topography across the strait is studied based on a nonhydrostatic numerical model. The model captures most wave characteristics shown by satellite data. A typical inter-packets distance on the Barents Sea side is about 25 km in summer, with a phase speed about 0.65 m/s. A northward background current may intensify the accumulation of energy during generation, but it has little influence on the other properties of the generated waves. The single internal solitary wave(ISW) structure is a special phenomenon that follows major wave trains, with a distance about 5–8 km. This wave is generated with the leading wave packets during the same tidal period. When a steady current toward the Kara Sea is included, the basic generation process is similar, but the waves toward the Kara Sea weaken and display an internal bore-like structure with smaller amplitude than in the control experiment. In winter, due to the growth of sea ice, stratification across the Kara Strait is mainly determined by the salinity, with an almost uniform temperature close to freezing. A pycnocline deepens near the middle of the water depth(Barents Sea side), and the NIWs process is not as important as the NIWs process in summer. There is no fission process during the simulation.

Propagation and modulational instability of Rossby waves in stratified fluids

Perturbation analysis and scale expansion are used to derive the(2+1)-dimensional coupled nonlinear Schr¨odinger(CNLS)equations that can describe interactions of two Rossby waves propagating in stratified fluids.The(2+1)-dimensional equations can reflect and describe the wave propagation more intuitively and accurately.The properties of the two waves in the process of propagation can be analyzed by the solution obtained from the equations using the Hirota bilinear method,and the influence factors of modulational instability are analyzed.The results suggest that,when two Rossby waves with slightly different wave numbers propagate in the stratified fluids,the intensity of bright soliton decreases with the increases of dark soliton coefficients.In addition,the size of modulational instable area is related to the amplitude and wave number in y direction.

Partial extinction, permanence and global attractivity innonautonomous n-species Lotka-Volterra competitivesystems with impulses

In this paper, the qualitative properties of general nonautonomous Lotka-Volterran-species competitive systems with impulsive e？ects are studied. Some new criteria on thepermanence, extinction and global attractivity of partial species are established by used themethods of inequalities estimate and Liapunov functions. As applications, nonautonomous twospecies Lotka-Volterra systems with impulses are discussed.

Magnetohydrodynamics hemodynamics hybrid nanofluid flow through inclined stenotic artery

The present study aims to perform computational simulations of twodimensional(2D)hemodynamics of unsteady blood flow via an inclined overlapping stenosed artery employing the Casson fluid model to discuss the hemorheological properties in the arterial region.A uniform magnetic field is applied to the blood flow in the radial direction as the magneto-hemodynamics effect is considered.The entropy generation is discussed using the second law of thermodynamics.The influence of different shape parameters is explored,which are assumed to have varied shapes(spherical,brick,cylindrical,platelet,and blade).The Crank-Nicolson scheme solves the equations and boundary conditions governing the flow.For a given critical height of the stenosis,the key hemodynamic variables such as velocity,wall shear stress(WSS),temperature,flow rate,and heat transfer coefficient are computed.

Automatic segmentation of gas plumes from multibeam water column images using a U-shape network

Cold seeps are widely developed on the seabed of continental margins and can form gas plumes due to the upward migration of methane-rich fluids.The detection and automatic segmentation of gas plumes are of great significance in locating and studying the cold seep system that is usually accompanied by hydrate layers in the subsurface.A multibeam echo-sounder system(MBES)can record the complete backscatter intensity of the water column,and it is one of the most effective means for detecting cold seeps.However,the gas plumes recorded in multibeam water column images(WCI)are usually blurred due to the interference of the complicated water environment and the sidelobes of the MBES,making it difficult to obtain the effective segmentation.Therefore,based on the existing UNet semantic segmentation network,this paper proposes an AP-UNet network combining the convolutional block attention module and the pyramid pooling module for the automatic segmentation and extraction of gas plumes.Comparative experiments are conducted among three traditional segmentation methods and two deep learning methods.The results show that the AP-UNet segmentation model can effectively suppress complicated water column noise interference.The segmentation precision,the Dice coefficient,and the recall rate of this model are 92.09%,92.00%,and 92.49%,respectively,which are 1.17%,2.10%,and 2.07%higher than the results of the UNet.

A Model of Perfect Pediatric Vaccination of Dengue with Delay and Optimal Control

A delayed mathematical model of Dengue dynamical transmission between vector mosquitoes and human, incorporating a control strategy of perfect pediatric vaccination is proposed in this paper. By some analytical skills, we obtain the existence of disease-free equilibria and endemic equilibrium, the necessary conditions of global asymptotical stability about two disease-free equilibria. Further, by Pontryagin’s maximum principle, we obtain the optimal control of the disease. Finally, numerical simulations are carried out to verify the correctness of the theoretical results and feasibility of the control measure.

THE SASA-SATSUMA EQUATION ON A NON-ZERO BACKGROUND: THE INVERSE SCATTERING TRANSFORM AND MULTI-SOLITON SOLUTIONS

We concentrate on the inverse scattering transformation for the Sasa-Satsuma equation with 3×3 matrix spectrum problem and a nonzero boundary condition. To circumvent the multi-value of eigenvalues, we introduce a suitable two-sheet Riemann surface to map the original spectral parameter k into a single-valued parameter z. The analyticity of the Jost eigenfunctions and scattering coefficients of the Lax pair for the Sasa-Satsuma equation are analyzed in detail. According to the analyticity of the eigenfunctions and the scattering coefficients, the z-complex plane is divided into four analytic regions of D_(j) : j = 1, 2, 3, 4. Since the second column of Jost eigenfunctions is analytic in D_(j), but in the upper-half or lowerhalf plane, we introduce certain auxiliary eigenfunctions which are necessary for deriving the analytic eigenfunctions in Dj. We find that the eigenfunctions, the scattering coefficients and the auxiliary eigenfunctions all possess three kinds of symmetries;these characterize the distribution of the discrete spectrum. The asymptotic behaviors of eigenfunctions, auxiliary eigenfunctions and scattering coefficients are also systematically derived. Then a matrix Riemann-Hilbert problem with four kinds of jump conditions associated with the problem of nonzero asymptotic boundary conditions is established, from this N-soliton solutions are obtained via the corresponding reconstruction formulae. The reflectionless soliton solutions are explicitly given. As an application of the N-soliton formula, we present three kinds of single-soliton solutions according to the distribution of discrete spectrum.

Rational solutions of Painlevé-Ⅱequation as Gram determinant

Under the Flaschka-Newell Lax pair,the Darboux transformation for the Painlevé-Ⅱequation is constructed by the limiting technique.With the aid of the Darboux transformation,the rational solutions are represented by the Gram determinant,and then we give the large y asymptotics of the determinant and the rational solutions.Finally,the solution of the corresponding Riemann-Hilbert problem is obtained from the Darboux matrices.

GLOBAL ASYMPTOTICAL PROPERTIES FOR A DIFFUSED HBV INFECTION MODEL WITH CTL IMMUNE RESPONSE AND NONLINEAR INCIDENCE

This article proposes a diffused hepatitis B virus （HBV） model with CTL immune response and nonlinear incidence for the control of viral infections. By means of different Lyapunov functions, the global asymptotical properties of the viral-free equilibrium and immune-free equilibrium of the model are obtained. Global stability of the positive equilibrium of the model is also considered. The results show that the free diffusion of the virus has no effect on the global stability of such HBV infection problem with Neumann homogeneous boundary conditions.

Soliton, breather, and rogue wave solutions for solving the nonlinear Schrodinger equation using a deep learning method with physical constraints

The nonlinear Schrodinger equation is a classical integrable equation which contains plenty of significant properties and occurs in many physical areas.However,due to the difficulty of solving this equation,in particular in high dimensions,lots of methods are proposed to effectively obtain different kinds of solutions,such as neural networks among others.Recently,a method where some underlying physical laws are embeded into a conventional neural network is proposed to uncover the equation’s dynamical behaviors from spatiotemporal data directly.Compared with traditional neural networks,this method can obtain remarkably accurate solution with extraordinarily less data.Meanwhile,this method also provides a better physical explanation and generalization.In this paper,based on the above method,we present an improved deep learning method to recover the soliton solutions,breather solution,and rogue wave solutions of the nonlinear Schrodinger equation.In particular,the dynamical behaviors and error analysis about the one-order and two-order rogue waves of nonlinear integrable equations are revealed by the deep neural network with physical constraints for the first time.Moreover,the effects of different numbers of initial points sampled,collocation points sampled,network layers,neurons per hidden layer on the one-order rogue wave dynamics of this equation have been considered with the help of the control variable way under the same initial and boundary conditions.Numerical experiments show that the dynamical behaviors of soliton solutions,breather solution,and rogue wave solutions of the integrable nonlinear Schrodinger equation can be well reconstructed by utilizing this physically-constrained deep learning method.

LUMP AND INTERACTION SOLUTIONS TO LINEAR (4+1)-DIMENSIONAL PDES

Taking a class of linear(4+1)-dimensional partial differential equations as examples, we would like to show that there exist lump solutions and interaction solutions in(4+1)-dimensions. We will compute abundant lump solutions and interaction solutions to the considered linear(4+1)-dimensional partial differential equations via symbolic computations,and plot three specific solutions with Maple plot tools, which supplements the existing literature on lump, rogue wave and breather solutions and their interaction solutions in soliton theory.

Anomalous reactivity of thermo-bioconvective nanofluid towards oxytactic microorganisms

The peristaltic flow of a non-Newtonian nanofluid with swimming oxytactic microorganisms through a space between two infinite coaxial conduits is investigated. A variable magnetic field is applied on the flow. The bioconvection flow and heat transfer in the porous annulus are formulated, and appropriate transformations are used, leading to the non-dimensionalized ruling partial differential equation model. The model is then solved by using the homotopy perturbation scheme. The effects of the germane parameters on the velocity profile, temperature distribution, concentration distribution, motile microorganism profile, oxytactic profile, pressure rise, and outer and inner tube friction forces for the blood clot and endoscopic effects are analyzed and presented graphically.It is noticed that the pressure rise and friction forces attain smaller values for the endoscopic model than for the blood clot model. The present analysis is believed to aid applications constituting hemodynamic structures playing indispensable roles inside the human body since some blood clotting disorders, e.g., haemophilia, occur when some blood constituents on the artery wall get confined away from the wall joining the circulation system.

Lump-type solutions of a generalized Kadomtsev–Petviashvili equation in(3+1)-dimensions

Through the Hirota bilinear formulation and the symbolic computation software Maple, we construct lump-type solutions for a generalized(3+1)-dimensional Kadomtsev-Petviashvili(KP) equation in three cases of the coefficients in the equation. Then the sufficient and necessary conditions to guarantee the analyticity of the resulting lump-type solutions(or the positivity of the corresponding quadratic solutions to the associated bilinear equation) are discussed. To illustrate the generality of the obtained solutions, two concrete lump-type solutions are explicitly presented, and to analyze the dynamic behaviors of the solutions specifically, the three-dimensional plots and contour profiles of these two lump-type solutions with particular choices of the involved free parameters are well displayed.

A new model for algebraic Rossby solitary waves in rotation fluid and its solution

A generalized Boussinesq equation that includes the dissipation effect is derived to describe a kind of algebraic Rossby solitary waves in a rotating fluid by employing perturbation expansions and stretching transformations of time and space.Using this equation, the conservation laws of algebraic Rossby solitary waves are discussed. It is found that the mass, the momentum, the energy, and the velocity of center of gravity of the algebraic solitary waves are conserved in the propagation process. Finally, the analytical solution of the equation is generated. Based on the analytical solution, the properties of the algebraic solitary waves and the dissipation effect are discussed. The results point out that, similar to classic solitary waves,the dissipation can cause the amplitude and the speed of solitary waves to decrease; however, unlike classic solitary waves,the algebraic solitary waves can split during propagation and the decrease of the detuning parameter can accelerate the occurrence of the solitary waves fission phenomenon.

Numerical Computations of Nonlocal Schrodinger Equations on the Real Line

The numerical computation of nonlocal Schrödinger equations (SEs) on the whole real axis is considered. Based on the artifcial boundary method, we frst derive the exact artifcial nonrefecting boundary conditions. For the numerical implementation, we employ the quadrature scheme proposed in Tian and Du (SIAM J Numer Anal 51:3458-3482, 2013) to discretize the nonlocal operator, and apply the z-transform to the discrete nonlocal system in an exterior domain, and derive an exact solution expression for the discrete system. This solution expression is referred to our exact nonrefecting boundary condition and leads us to reformulate the original infnite discrete system into an equivalent fnite discrete system. Meanwhile, the trapezoidal quadrature rule is introduced to discretize the contour integral involved in exact boundary conditions. Numerical examples are fnally provided to demonstrate the efectiveness of our approach.

Well-Posedness of an N-Unit Series System with Finite Number of Vacations

We investigate the solution of an N-unit series system with finite number of vacations. By using C0-semigroup theory of linear operators, we prove well-posedness and the existence of the unique positive dynamic solution of the system.

A note on “Lattice soliton equation hierarchy and associated properties”

We demonstrate that the new hierarchy of integrable lattice equations in Chin. Phys. B 21 090202(2012) can be changed into the integrable lattice hierarchy in Chin. Phys. B 13 1009(2004) by using a very simple transformation.