The Schrödinger–Langevin equation with and without thermal fluctuations

The Schrödinger–Langevin equation (SLE) is considered as an effective open quantum system formalism suitable for phenomenological applications involving a quantum subsystem interacting with a thermal bath. We focus on two open issues relative to its solutions: the stationarity of the excited states of the non-interacting subsystem when one considers the dissipation only and the thermal relaxation toward asymptotic distributions with the additional stochastic term. We first show that a proper application of the Madelung/polar transformation of the wave function leads to a non zero damping of the excited states of the quantum subsystem. We then study analytically and numerically the SLE ability to bring a quantum subsystem to the thermal equilibrium of statistical mechanics. To do so, concepts about statistical mixed states and quantum noises are discussed and a detailed analysis is carried with two kinds of noise and potential. We show that within our assumptions the use of the SLE as an effective open quantum system formalism is possible and discuss some of its limitations.     

The Monge–Ampère equation: Various forms and numerical solution

We present three novel forms of the Monge–Ampère equation, which is used, e.g., in image processing and in reconstruction of mass transportation in the primordial Universe. The central role in this paper is played by our Fourier integral form, for which we establish positivity and sharp bound properties of the kernels. This is the basis for the development of a new method for solving numerically the space-periodic Monge–Ampère problem in an odd-dimensional space. Convergence is illustrated for a test problem of cosmological type, in which a Gaussian distribution of matter is assumed in each localised object, and the right-hand side of the Monge–Ampère equation is a sum of such distributions.     

Decompositions of the Kadomtsev–Petviashvili equation and their symmetry reductions

Starting with a decomposition conjecture, we carefully explain the basic decompositions for the Kadomtsev–Petviashvili(KP) equation as well as the necessary calculation procedures, and it is shown that the KP equation allows the Burgers–STO(BSTO) decomposition, two types of reducible coupled BSTO decompositions and the BSTO–KdV decomposition. Furthermore, we concentrate ourselves on pointing out the main idea and result of B?cklund transformation of the KP equation based on a special superpositi...     

The Gibbons–Tsarev equation: symmetries,invariant solutions,and applications

In this paper we present the full classification of symmetry-invariant solutions for the Gibbons–Tsarev equation. Then we use these solutions to construct explicit expressions for two-component reductions of Benney’s moments equations, to get solutions of Pavlov’s equation, and to find integrable reductions of the Ferapontov–Huard–Zhang system, which describes implicit two-phase solutions of the dKP equation.     

The Schrödinger group in molecular quantum mechanics: beyond the Born–Oppenheimer approximation

The Galillei transformation invariant form of molecular quantum mechanics is obtained, which is not restricted by the Born–Oppenheimer approximation. The molecular Hamiltonian takes then the form of a linear combination of operators of the Schrödinger group, which define the new space–time molecular symmetry properties (e.g. helicity).

The puzzling homochirality of the hydrogen bonded biomolecular systems appears then as a simple result of the molecular space-time symmetry.     

The Hirota bilinear method for the coupled Burgers equation and the high-order Boussinesq–Burgers equation

This paper studies the coupled Burgers equation and the high-order Boussinesq–Burgers equation. The Hirota bilinear method is applied to show that the two equations are completely integrable. Multiple-kink (soliton) solutions and multiple-singular-kink (soliton) solutions are derived for the two equations.     

Fermionic extension of the scalar Born–Infeld equation and its relation to the supersymmetric Chaplygin gas

In this Letter, we present a fermionic extension of the scalar Born–Infeld equation, which has been derived from the Nambu–Goto superstring action in (2+1)$(2+1)$ dimensions through the Cartesian parameterization. It is demonstrated that in the relativistic limit where c→∞$c\to \infty$, the fermionic Born–Infeld model reduces to the supersymmetric Chaplygin gas model in one spatial dimension. However, the supersymmetry itself is not preserved.     

Analysis of the seventh-order Caputo fractional KdV equation: applications to the Sawada–Kotera–Ito and Lax equations

In this study, we investigate the seventh-order nonlinear Caputo time-fractional KdV equation.The suggested model’s solutions, which have a series form, are obtained using the hybrid ZZtransform under the aforementioned fractional operator. The proposed approach combines the homotopy perturbation method(HPM) and the ZZ-transform. We consider two specific examples with suitable initial conditions and find the series solution to test their applicability. To demonstrate the utility of the presented...     

Dispersive traveling wave solutions to the space–time fractional equal-width dynamical equation and its applications

In this article, some new traveling wave solutions to the space–time fractional equal-width equation are constructed with the help of the extended Fan sub-equation method. A simple transformation is introduced to convert the fractional order partial differential equation into an ordinary differential equation. As a result, the bright, dark, singular and combined wave solitons are observed for different values of two parameters. Moreover, the graphical representations are also depicted.     

A novel secular equation for the coupled-band envelope-function approximation for superlattices and application to InAs/InxGa1 − xSb superlattices

This paper derives and demonstrates a new secular equation for the coupled-band envelope-function approximation (EFA) formalism for superlattices in order to overcome the difficulty of handling rapidly growing or decaying wavefunction components, in particular, the ‘wing solutions’. In a second development, the generally nonHermitian secular equation is made Hermitian, making it easier to locate multiply degenerate roots. In the process, the simple Kronig–Penney model is recast into a form closely related to that for the underlying quantum well (QW) problem. The present method is applicable to Burt's EFA formalism. An InAs/InGaSb type-II superlattice is used to demonstrate the method.     

On the solutions and the steady states of a master equation

A complete characterization of the time behavior of the means and variance of a stochastic process which is generated by a finite number of independent systems is presented based on the master equation for the conditional probability. It is found that the means and variance relax to a steady state and that the steady state will be independent of the initial state if and only if a matrix related to the transition matrix is nonsingular. Finally, the result that the variance approaches its steady-state form at twice the rate of the means is shown to depend on the nonsingularity of the same matrix.     

The difference between Schrödinger equation derived from Schrödinger map and Landau-Lifshitz equation

We provide an explicit blow up solution of Schrödinger equation derived from Schrödinger map. Consequently we show the non-equivalence between the Schrödinger equation and Landau-Lifshitz equation. We also find that two class of equivariant solutions of Landau-Lifshitz equation are static.     

An energy-transport model for semiconductors derived from the Boltzmann equation

An energy-transport model is rigorously derived from the Boltzmann transport equation of semiconductors under the hypothesis that the energy gain or loss of the electrons by the phonon collisions is weak. Retaining at leading order electron-electron collisions and elastic collisions (i.e., impurity scattering and the elastic part of phonon collisions), a rigorous diffusion limit of the Boltzmann equation can be carried over, which leads to a set of diffusion equations for the electron density and temperature. The derivation is given in both the degenerate and nondegenerate cases.     

The double Caldeira-Leggett model: Derivation and solutions of the master equations, reservoir-induced interactions and decoherence

In this paper we analyze the double Caldeira-Leggett model: the path integral approach to two interacting dissipative harmonic oscillators. Assuming a general form of the interaction between the oscillators, we consider two different situations: (i) when each oscillator is coupled to its own reservoir, and (ii) when both oscillators are coupled to a common reservoir. After deriving and solving the master equation for each case, we analyze the decoherence process of particular entanglements in the positional space of both oscillators. To analyze the decoherence mechanism we have derived a general decay function, for the off-diagonal peaks of the density matrix, which applies both to common and separate reservoirs. We have also identified the expected interaction between the two dissipative oscillators induced by their common reservoir. Such a reservoir-induced interaction, which gives rise to interesting collective damping effects, such as the emergence of relaxation- and decoherence-free subspaces, is shown to be blurred by the high-temperature regime considered in this study. However, we find that different interactions between the dissipative oscillators, described by rotating or counter-rotating terms, result in different decay rates for the interference terms of the density matrix.     

The population and decay evolution of a qubit under the time-convolutionless master equation

We consider the population and decay of a qubit under the electromagnetic environment. Employing the time-convolutionless master equation, we investigate the Markovian and non-Markovian behaviour of the corresponding perturbation expansion. The Jaynes-Cummings model on resonance is investigated. Some figures clearly show the different evolution behaviours. The reasons are interpreted in the paper.     

The Fokker-Planck equation in the second-order pitch angle approximation and its exact solution

The diffusive particle propagation and its pitch angle scattering is studied using kinetic equation of the Fokker-Planck form. The case is considered when charged particles preferable propagate along the strong mean magnetic field direction and undergo the pitch angle scattering with respect to it. The paper deals with solution of the equation for particle distribution function in the second-order approximation in the pitch angle. The exact analytical solution is obtained in an integral form. The well-known solution in the first-order pitch angle approximation can be restored performing the small time limit in the result. Unlike the first-order solution the obtained solution in the second approximation rightly shows that the pitch angle diffusion is closely connected with the particle transport along the mean magnetic field. The expression for particle density for the point instantaneous unidirectional source also has been obtained.     

A three-dimensional finite elements approach for the coupled radiative transfer equation and diffusion approximation modeling in fluorescence imaging

During the last few years a quite large number of fluorescence molecular imaging applications have been reported in the literature, as one of the most challenging aspects in medical imaging is to “see” a tumor embedded into tissue, which is a turbid medium, by using fluorescent probes for tumor labeling. However, the forward solvers, required for the successful convergence of the inverse problem, are still lacking accuracy and time feasibility. Moreover, initialization of these solvers may be proven even more difficult than solving the inverse problem itself. This paper describes in depth a coupled radiative transfer equation and diffusion approximation model for solving the forward problem in fluorescence imaging. The theoretical confrontation of these solvers comprises the model deployment, its Galerkin finite elements approximation and the domain discretization scheme. Finally, a new optical properties mapping algorithm, based on super-ellipsoid models, is implemented, providing a fully automated simulation target construction within feasible time.     

A note on surface stress and surface tension and their interrelation via Shuttleworth’s equation and the Lippmann equation

Some aspects of the thermodynamics of solid surfaces, in particular with respect to the surface stress, f, and surface tension, γ, including the case of solid electrodes, are examined in view of their controversial discussion in part of the recent literature. By inspection of the phenomenology that requires a distinction between f and γ, and of a toy model designed to highlight the underlying fundamental science, it is shown that some of the recent publications give misleading conclusions. These include [V.A. Marichev, Surf. Sci. 600 (19) (2006) 4527; E.M. Gutman, J. Phys. Condens. Matter. 7 (48) (1995) L663; D.J. Bottomley, T. Ogino, Phys. Rev. B 63 (2001) 165412]. In spite of claims to the contrary, the validity of the equations of Shuttleworth, Lippmann, and Couchman and Davidson is not impaired by the arguments of the aforementioned articles.     

Generalized heat diffusion equations with variable coefficients and their fractalization from the Black-Scholes equation

In this study, we prove that modified diffusion equations, including the generalized Burgers' equation with variable coefficients, can be derived from the Black-Scholes equation with a time-dependent parameter based on the propagator method known in quantum and statistical physics.The extension for the case of a local fractal derivative is also addressed and analyzed.     

Bounded multi-soliton solutions and their asymptotic analysis for the reversal-time nonlocal nonlinear Schrödinger equation

In this paper, we construct the Darboux transformation (DT) for the reverse-time integrable nonlocal nonlinear Schrödinger equation by loop group method. Then we utilize the DT to derive soliton solutions with zero seed. We investigate the dynamical properties for those solutions and present a sufficient condition for the non-singularity of multi-soliton solutions. Furthermore, the asymptotic analysis of bounded multi-solutions has also been established by the determinant formula.