Extreme waves statistics for the Ablowitz-Ladik system

We examine statistics of waves for the problem of modulation instability development in the framework of discrete integrable Ablowitz-Ladik (AL) system. Modulation instability depends on one free parameter h that has the meaning of the coupling between the nodes on the lattice. For strong coupling h ? 1, the probability density functions (PDFs) for waves amplitudes coincide with that for the continuous classical nonlinear Schrödinger equation; the PDFs for both systems are very close to Rayleigh ones. When the coupling is weak h ～ 1, there appear highly localized waves with very large amplitudes, that drastically change the PDFs to significantly non-Rayleigh ones, with so-called “fat tails” when the probability of a large wave occurrence is by several orders of magnitude higher than that predicted by the linear theory. Evolution of amplitudes for such rogue waves with time is similar to that of the Peregrine solution for the classical nonlinear Schrödinger equation.     

A New Approach to the Exact Solutions of the Effective Mass Schrödinger Equation

Effective mass Schrödinger equation is solved exactly for a given potential. Nikiforov-Uvarov method is used to obtain energy eigenvalues and the corresponding wave functions. A free parameter is used in the transformation of the wave function. The effective mass Schrödinger equation is also solved for the Morse potential transforming to the constant mass Schrödinger equation for a potential. One can also get solution of the effective mass Schrödinger equation starting from the constant mass Schrödinger equation.     

Stochastic electrodynamics. I. On the stochastic zero-point field

This is the first in a series of papers that present a new classical statistical treatment of the system of a charged harmonic oscillator (HO) immersed in an omnipresent stochastic zero-point (ZP) electromagnetic radiation field. This paper establishes the Gaussian statistical properties of this ZP field using Bourret's postulate that all statistical moments of the stochastic field plane waves at a given space-time point should agree with their corresponding quantized field vacuum expectations. This postulate is more than adequate to derive the Planck spectrum classically via Boyer's and Theimer's methods, but it requires that the stochastic amplitude of each linearly polarized plane wave in the field contain two independent Gaussian random variables, not just a random phase as has sometimes been assumed. In the succeeding papers in the series, the total motion of a charged HO is described by a fully renormalized dipole-approximation Abraham-Lorentz equation. This leads without further approximation to the following major results concerning this stochastic electrodynamics (SED) of the HO: i) The ensemble-average Liouville equation for the oscillator-ZP field system in the presence of an arbitrary applied classical radiation field is exactly equivalent to the usual time-dependent Schrödinger equation supplemented by an explicit radiation reaction vector potential similar to that of the Crisp-Jaynes-Stroud theory; ii) this SED Schrödinger equation for the HO is incomplete, insmuch as there exists a companion equation that restricts initial conditions such that the corresponding Wigner phase-space distribution is always positive; iii) the wave function of the SED Schrödinger equation has thea priori significance of position probability amplitude; iv) first-order transition rates predicted for the HO by this theory agree with those predicted by quantum electrodynamics for resonance absorption and spontaneous emission, which occurs with no triggering necessary; and v) if SED is taken seriously, then the concepts of quantized energies and photons must be abandoned.     

A numerical method for fractional Schrödinger equation of Lennard-Jones potential

Depending on fractional analysis, we find a numerical algorithm to solve the time-independent fractional Schrödinger equation in case of Lennard-Jones potential in one dimension. We apply the algorithm for multiple values of the fractional parameter of the space-dependent fractional Schrödinger equation and multiple values of the system's energy to find the wave function and the probability in these cases.     

A perfectly matched layer approach to the nonlinear Schrödinger wave equations

Absorbing boundary conditions (ABCs) are generally required for simulating waves in unbounded domains. As one of those approaches for designing ABCs, perfectly matched layer (PML) has achieved great success for both linear and nonlinear wave equations. In this paper we apply PML to the nonlinear Schrödinger wave equations. The idea involved is stimulated by the good performance of PML for the linear Schrödinger equation with constant potentials, together with the time-transverse invariant property held by the nonlinear Schrödinger wave equations. Numerical tests demonstrate the effectiveness of our PML approach for both nonlinear Schrödinger equations and some Schrödinger-coupled systems in each spatial dimension.     

New optical soliton solutions for nonlinear complex fractional Schrödinger equation via new auxiliary equation method and novel $$({G'}/{G})$$-expansion method

In this research, we apply two different techniques on nonlinear complex fractional nonlinear Schrödinger equation which is a very important model in fractional quantum mechanics. Nonlinear Schrödinger equation is one of the basic models in fibre optics and many other branches of science. We use the conformable fractional derivative to transfer the nonlinear real integer-order nonlinear Schrödinger equation to nonlinear complex fractional nonlinear Schrödinger equation. We apply new auxiliary equation method and novel \(\left( {G'}/{G}\right) \)-expansion method on nonlinear complex fractional Schrödinger equation to obtain new optical forms of solitary travelling wave solutions. We find many new optical solitary travelling wave solutions for this model. These solutions are obtained precisely and efficiency of the method can be demonstrated.     

Bifurcations and solutions for the generalized nonlinear Schrödinger equation

The theory of bifurcations for dynamical system is employed to construct new exact solutions of the generalized nonlinear Schrödinger equation. Firstly, the generalized nonlinear Schrödinger equation was converted into ordinary differential equation system by using traveling wave transform. Then, the system's Hamiltonian, orbits phases diagrams are found. Finally, six families of solutions are constructed by integrating along difference orbits, which consist of Jacobi elliptic function solutions, hyperbolic function solutions, trigonometric function solutions, solitary wave solutions, breaking wave solutions, and kink wave solutions.     

Entanglement in the Schrödinger experiment

The mixed and entanglement states have been analyzed in the Schrödinger experiment. It is known that, in an open system, the “Schrödinger cat” paradox is explained by the decoherence phenomenon, but, in a closed system, it is explained by the Everett-Wheeler many-world interpretation of quantum mechanics. The quantum real world can be presented as a complex multispatial geometric figure and the classical world is one of the faces of this figure. In this paper it is shown that this figure is the simplex that is well known in the functional analysis. Such an interpretation of quantum mechanics enables one to obtain the nonuniform wave equation, and the Schrödinger equation is the uniform equation of this one. Perhaps this equation is the equation of the subquantum world about which Einstein has written.     

Conformal covariance and the probability interpretation of wave equations

The Dirac and the Schrödinger fields are singled out among all other fields by their conformal dimension $\text{l = ?}\text{3}\text{2}$, a necessary condition for the field to have also a probability interpretation as a wave function. The correct dimension of the Schrödinger field comes about by an intricate contraction of the conformal group to the Schrödinger group.     

A General Approach for the Exact Solution of the Schrödinger Equation

The Schrödinger equation is solved exactly for some well known potentials. Solutions are obtained reducing the Schrödinger equation into a second order differential equation by using an appropriate coordinate transformation. The Nikiforov-Uvarov method is used in the calculations to get energy eigenvalues and the corresponding wave functions.     

Quasi-molecular states of neutrons in dinuclear systems and their effect on fusion of heavy atomic nuclei

The evolution of the wave functions of valence neutrons upon collisions of heavy nuclei having energies near the Coulomb barrier has been investigated by numerical solution of the multidimensional time-dependent Schrödinger equation. It is shown that even before overcoming the barrier by nuclei the wave functions of valence neutrons are extended to their volumes with occupation of quasi-molecular states. This process results in a significant effect of neutron transfers on nuclear fusion and, in particular, in a significant increase in the probability of subbarrier fusion for certain combinations of nuclei.     

Bayesian approach to electron correlation in density functional theory

In the present communication we applied the Bayesian conditional probability approach to the wave function of a many‐electron system that resulted in the appearance of a quantum vector potential in the DFT Schrödinger equation due to electron correlation, apart from the correlation energy term. Mathematically, the effect of this vector potential is equivalent to a magnetic field that corresponds in particular to a conservative irrotational one if it is considered in connection with the correlation potential. An analysis of the effect of the correlation momentum on the electronic transitions suggested that the electron correlation increases the transition probability.     

Bi-Confluent Heun Potentials for a Stationary Relativistic Wave Equation for a Spinless Particle

The variety of bi-confluent Heun potentials for a stationary relativistic wave equation for a spinless particle is presented. The physical potentials and energy spectrum of this wave equation are related to those for a corresponding Schrödinger equation in the sense that all the potentials derived for the latter equation are also applicable for the wave equation under consideration. We show that in contrast to the Schrödinger equation the characteristic spatial length of the potential imposes a restriction on the energy spectrum that directly reflects the uncertainty principle. Studying the inversesquare- root bi-confluent Heun potential, it is shown that the uncertainty principle limits, from below, the principal quantum number for the bound states, i.e., physically feasible states have an infimum cut so that the ground state adopts a higher quantum number as compared to the Schrödinger case.     

Theory of Stochastic Schrödinger Equation in Complex Vector Space

A generalized Schrödinger equation containing correction terms to classical kinetic energy, has been derived in the complex vector space by considering an extended particle structure in stochastic electrodynamics with spin. The correction terms are obtained by considering the internal complex structure of the particle which is a consequence of stochastic average of particle oscillations in the zeropoint field. Hence, the generalised Schrödinger equation may be called stochastic Schrödinger equation. It is found that the second order correction terms are similar to corresponding relativistic corrections. When higher order correction terms are neglected, the stochastic Schrödinger equation reduces to normal Schrödinger equation. It is found that the Schrödinger equation contains an internal structure in disguise and that can be revealed in the form of internal kinetic energy. The internal kinetic energy is found to be equal to the quantum potential obtained in the Madelung fluid theory or Bohm statistical theory. In the rest frame of the particle, the stochastic Schrödinger equation reduces to a Dirac type equation and its Lorentz boost gives the Dirac equation. Finally, the relativistic Klein–Gordon equation is derived by squaring the stochastic Schrödinger equation. The theory elucidates a logical understanding of classical approach to quantum mechanical foundations.     

Nonlinear electrohydrodynamic Rayleigh-Taylor instability with mass and heat transfer subject to a vertical oscillating force and a horizontal electric field

Weakly nonlinear stability of interfacial waves propagating between two electrified inviscid fluids influenced by a vertical periodic forcing and a constant horizontal electric field is studied. Based on the method of multiple-scale expansion for a small-amplitude periodic force, two parametric nonlinear Schrödinger equations with complex coefficients are derived in the resonance cases. A standard nonlinear Schrödinger equation with complex coefficients is derived in the nonresonance case. A temporal solution is carried out for the parametric nonlinear Schrödinger equation. The stability analysis is discussed both analytically and numerically.     

On the Linear Forms of the Schrödinger Equation

Generalising the linearisation procedure used by Dirac and later by Lévy-Leblond, we derive the first-order non-relativistic wave equations for particles of spin 1 and spin 3/2 starting from the Schrödinger equation. By the introduction in the momentum of a correction linear in coordinates, we establish the wave equation of the radial harmonic oscillator with spin-orbit coupling.     

Thermal mechanics: A quantum mechanical analogue of nonequilibrium statistical thermodynamics

A formal but not conventional equivalence between stochastic processes in nonequilibrium statistical thermodynamics and Schrödinger dynamics in quantum mechanics is shown. It is found, for each stochastic process described by a stochastic differential equation of Itô type, there exists a Schrödinger-like dynamics in which the absolute square of a wavefunction gives us the same probability distribution as the original stochastic process. In utilizing this equivalence between them, that is, rewriting the stochastic differential equation by an equivalent Schrödinger equation, it is possible to obtain the notion of deterministic limit of the stochastic process as a semi-classical limit of the “Schrödinger” equation. The deterministic limit thus obtained improves the conventional deterministic approximation in the sense of Onsager-Machlup. The present approach is valid for a general class of stochastic equations where local drifts and diffusion coefficients depend on the position. Two concrete examples are given. It should be noticed that the approach in the present form has nothing to do with the conventional one where only a formal similarity between the Fokker-Planck equation and the Schrödinger equation is considered.     

Long Time Energy Transfer in the Random Schrödinger Equation

We consider the long time behavior of solutions of the d-dimensional linear Boltzmann equation that arises in the weak coupling limit for the Schrödinger equation with a time-dependent random potential. We show that the intermediate mesoscopic time limit satisfies a Fokker–Planck type equation with the wave vector performing a Brownian motion on the (d ? 1)-dimensional sphere of constant energy, as in the case of a time-independent Schrödinger equation. However, the long time limit of the solution with an isotropic initial data satisfies an equation corresponding to the energy being the square root of a Bessel process of dimension d/2.