Quantum Effects of Mesoscopic Inductance and Capacity Coupling Circuits

Using the quantum theory for a mesoscopic circuit based on the discretenes of electric charges, the finite-difference Schrödinger equation of the non-dissipative mesoscopic inductance and capacity coupling circuit is achieved. The Coulomb blockade effect, which is caused by the discreteness of electric charges, is studied. Appropriately choose the components in the circuits, the finite-difference Schrödinger equation can be divided into two Mathieu equations in \hat p representation. With the WKBJ method, the currents quantum fluctuations in the ground states of the two circuits are calculated. The results show that the currents quantum zero-point fluctuations of the two circuits are exist and correlated.     

Macroscopic Quantum Coherent Phenomena in the Mesoscopic Electric Circuit

The quantum theory for mesoscopic electric circuit with charge discreteness is briefly described. The Schrödinger equation of the mesoscopic electric circuit with external source which is the time function has been proposed. By using the instanton methods, the macroscopic quantum coherent phenomena and effective capacitance oscillation in the mesoscopic electric circuit have been addressed.     

Bloch wave oscillation and Coulomb blockade in mesoscopic electric circuits

The quantum theory for mesoscopic electric circuits with charge discreteness is briefly described. The Schrödinger equation of the mesoscopic electric circuit with an external source which is the time function has been proposed. The Bloch wave oscillation and Coulomb blockade in the mesoscopic electric circuit have been addressed.     

Series Solution for Localization and Entanglement of an Exciton in a Quantum Dot Molecule by an ac Electric Field

The Schrödinger equation involving the phenomenon of the localization and entanglement for an exciton in a quantum dot molecule by an ac electric field is analytically investigated. New exact series solutions for the Schrödinger equation have been obtained for the first time. The analytical expressions can further describe the dynamical behaviors of an interacting electron-hole pair in a double coupled quantum dot molecule under an ac electric field accurately.     

Quantum Effect in a Diode Included Nonlinear Inductance-Capacitance Mesoscopic Circuit

The mesoscopic nonlinearinductance-capacitance circuit is a typical anharmonicoscillator, due to diodes included in the circuit. In this paper, using the advanced quantum theory of mesoscopic circuits, which based on the fundamental fact that the electric charge takes discrete value, the diode included mesoscopic circuit is firstly studied. Schrödinger equation of the system is a four-order difference equation in \hat{p} representation.Using the extended perturbative method, the detail energy spectrumand wave functions are obtained and verified, as an application ofthe results, the current quantum fluctuation in the ground state iscalculated. Diode is a basis component in a circuit, its quantization would popularize the quantum theory of mesoscopic circuits. The methods to solve the high order difference equation are helpful to the application of mesoscopic quantum theory.     

电荷不连续时电容耦合介观电路的量子回路方程及其能谱

考虑电荷具有不连续性的事实对双LC介观电路进行量子化,给出耦合形式的量子回路方程以及无相互作用Hamilton本征基矢下的电路能谱.结果表明,计及电荷离散性将使回路方程的形式发生明显变化;介观电路的能谱除与电路参数相关外,还明显地依赖于电荷的量子化性质.     

A Unified and Explicit Construction of N-Soliton Solutions for the Nonlinear Schrödinger Equation

An explicit N-fold Darboux transformation with multiparameters for nonlinear Schrödinger equation is constructed with the help of its Lax pairs and a reduction technique. According to this Darboux transformation, the solutions of the nonlinear Schrödinger equation are reduced to solving a linear algebraic system, from which a unified and explicit formulation of N-soliton solutions with multiparameters for the nonlinear Schrödinger equation is given.     

An Integrable Decomposition of Defocusing Nonlinear Schrödinger Equation

The method of nonlinearization of spectral problems is developed to thedefocusing nonlinear Schrödinger equation. As an application, an integrable decomposition of the defocusing nonlinear Schrödinger equation is presented.     

Lax Pairs and Integrability Conditions of Higher-Order Nonlinear Schrödinger Equations

We derive the Lax pairs and integrability conditions of the nonlinear Schrödinger equation with higher-order terms, complex potentials, and time-dependent coefficients. Cubic and quintic nonlinearities together with derivative terms are considered. The Lax pairs and integrability conditions for some of the well-known nonlinear Schrödinger equations, including a new equation which was not considered previously in the literature, are then derived as special cases. We show most clearly with a similarity transformation that the higher-order terms restrict the integrability to linear potential in contrast with quadratic potential for the standard nonlinear Schrödinger equation.     

An Integrable Discrete Generalized Nonlinear Schrödinger Equation and Its Reductions

An integrable discrete system obtained by the algebraization of the difference operator is studied. The system is named discrete generalized nonlinear Schrödinger (GNLS) equation, which can be reduced to classical discrete nonlinear Schrödinger (NLS) equation. Furthermore, all of the linear reductions for the discrete GNLS equation are given through the theory of circulant matrices and the discrete NLS equation is obtained by one of the reductions. At the same time, the recursion operator and symmetries of continuous GNLS equation are successfully recovered by its corresponding discrete ones.     

Mixed Local-Nonlocal Vector Schrödinger Equations and Their Breather Solutions

To the best of our knowledge, all nonlinearities in the known nonlinear integrable systems are either local or nonlocal. A natural problem is whether there exist some nonlinear integrable systems with both local and nonlocal nonlinearities, and how to solve this kinds of spectral nonlinear integrable systems with both local and nonlocal nonlinearities. Recently, some novel mixed local-nonlocal vector Schrödinger equations are presented, which are different from the single local and nonlocal coupled Schrödinger equation. We investigate the Darboux transformation of mixed local-nonlocal vector Schrödinger equations with a spectral problem. Starting from a special Lax pairs, the mixed localnonlocal vector Schrödinger equations are constructed. We obtain the one- and two- and N-soliton solution formulas of the mixed local-nonlocal vector Schrödinger equations with N-fold Darboux transformation. Based on the obtained solutions, the propagation and interaction structures of these multi-solitons are shown, the evolution structures of the one-solitons are exhibited, the overtaking elastic interactions among the two-breather solitons are considered. We find that unlike the local and nonlocal cases, the mixed local-nonlocal vector Schrödinger equations have some novel results. The results in this paper might be helpful for understanding some physical phenomena described in plasmas.     

Evolution of Arbitrary States under Fock-Darwin Hamiltonian and a Time-Dependent Electric Field

The method of path integral is employed to calculate the time evolution of the eigenstates of a charged particle under the Fock-Darwin(FD) Hamiltonian subjected to a time-dependent electric field in the plane of the system.An exact analytical expression is established for the evolution of the eigenstates.This result then provides a general solution to the time-dependent Schro¨dinger equation.     

On Differential Equations Describing 3-Dimensional Hyperbolic Spaces

In this paper, we introduce the notion of a (2+1)-dimensional differential equation describing three-dimensional hyperbolic spaces (3-h.s.). The (2+1)-dimensional coupled nonlinear Schrödinger equation and its sister equation, the (2+1)-dimensional coupled derivative nonlinear Schrödinger equation, are shown to describe 3-h.s. The (2+1)-dimensional generalized HF model: S_t={(1/2i)[S,S_y]+2iσS}_x, σ_x=-(1/4i)tr(SS_xS_y), in which S∈[GL_C(2)]/[GL_C(1)×GL_C(1)], provides another example of (2+1)-dimensional differential equations describing 3-h.s. As a direct consequence, the geometric construction of an infinite number of conservation laws of such equations is illustrated. Furthermore we display a new infinite number of conservation laws of the (2+1)-dimensional nonlinear Schrödinger equation and the (2+1)-dimensional derivative nonlinear Schrödinger equation by a geometric way.     

An Extended Subequation Rational Expansion Method and Solutions of (2+1)-Dimensional Cubic Nonlinear Schrödinger Equation

An extended subequation rational expansion method is presented and used to construct some exact analytical solutions of the (2+1)-dimensional cubic nonlinear Schrödinger equation. From our results, many known solutions of the (2+1)-dimensional cubic nonlinear Schrödinger equation can be recovered by means of some suitable selections of the arbitrary functions and arbitrary constants. With computer simulation, the properties of new non-travelling wave and coefficient function's soliton-like solutions, and elliptic solutions are demonstrated by some plots.     

Wronskian Approach and One-Dimensional Schrödinger Equation with Double-Well Potential

A Wronskian determinant approach is suggested to study the energy and the wave function for one-dimensional Schrödinger equation. An integral equation and its corresponding Green function are constructed. As an example, we employed this approach to study the problem of double-well potential with strong coupling. A series of expansion of ground state energy up to the second order approximation of iterative procedure is given.     

The Quantum Fluctuations of Two Coupled Josephson Junctions with the Discreteness of Electric Charge

The quantization of two Josephson junctions coupled via inductor with the discreteness of electric charges is proposed. The finite-difference Schrodinger equation of the circuit system has been obtained in charge representation, and the Schrodinger equation is turned to be Mathieu equation in flux representation. The wavefunction and energy spectrum can be solved by adopting the canonical transformation and unitary transformation method. The results indicate that the quantum fluctuations of the flux in the ground states of each mesh exist and are interrelated.     

Application of asymptotic iteration method to a deformed well problem

The asymptotic iteration method(AIM) is used to obtain the quasi-exact solutions of the Schr o¨dinger equation with a deformed well potential. For arbitrary potential parameters, a numerical aspect of AIM is also applied to obtain highly accurate energy eigenvalues. Additionally, the perturbation expansion, based on the AIM approach, is utilized to obtain simple analytic expressions for the energy eigenvalues.     

Josephson-Like Effects in the Mesoscopic Electric Circuit

Abstract The quantum theory for mesoscopic electric circuit with charge discreteness is briefly described. The Schrodinger equation of the mesoscopic electric circuit with external source which is a time function has been proposed. The Josephson-like effects in the mesoscopic electric circuit have been addressed.     

Multiple-pole solutions and degeneration of breather solutions to the focusing nonlinear Schrödinger equation

Based on the Hirota’s method, the multiple-pole solutions of the focusing Schrödinger equation are derived directly by introducing some new ingenious limit methods. We have carefully investigated these multi-pole solutions from three perspectives: rigorous mathematical expressions, vivid images, and asymptotic behavior. Moreover, there are two kinds of interactions between multiple-pole solutions: when two multiple-pole solutions have different velocities, they will collide for a short time; when two multiple-pole solutions have very close velocities, a long time coupling will occur. The last important point is that this method of obtaining multiple-pole solutions can also be used to derive the degeneration of N-breather solutions. The method mentioned in this paper can be extended to the derivative Schrödinger equation, Sine-Gorden equation, mKdV equation and so on.