1,045 Results Found

By applying Clark’s theorem as altered by Liu and Wang and the truncation method, we obtain a sequence of solutions for a Schrödinger–Poisson system $\left\{\begin{array}{cc}-\Delta u+V\left(x\right)u+\varphi u=f\left(u\right)\hfill & \mathrm{in}{\mathbb{R}}^3,\hfill \\ -\Delta \varphi =u^2\hfill & \mathrm{in}{\mathbb{R}}^3\hfill \end{array}\right.$ with negative ene...

In this paper, the cubic-quartic nonlinear Schrödinger and resonant nonlinear Schrödinger equation in parabolic law media are investigated to obtain the dark, singular, bright-singular combo and periodic soliton solutions. Two powerful meth...

Based on the assumption that the standard Schrödinger equation becomes gravitationally modified for massive macroscopic objects, two independent proposals have survived from the 1980s. The Schrödinger–Newton equation (1984) provides w...

This paper begins with the introduction of biquaternion algebra and its properties in compact, profound and comprehensible approach. The Schrödinger equation including both the scalar Aharonov-Bohm(sAB) and Aharonov-Casher(AC) effects, that are curre...

We establish an analogy between the Fokker–Planck equation describing evolutionary landscape dynamics and the Schrödinger equation which characterizes quantum mechanical particles, showing that a population with multiple genetic traits evo...

In flat spacetime, the Dirac equation is the “square root” of the Klein–Gordon equation in the sense that, by applying the square of the Dirac operator to the Dirac spinor, one recovers the equation duplicated for each component of...

For almost 75 years, the general solution for the Schrödinger equation was assumed to be generated by an exponential or a time-ordered exponential known as the Dyson series. We study the unitarity of a solution in the case of a singular Hamilton...

We analyze the generalized time-dependent Schrödinger equation for the force free case, as a generalization, for example, of the standard time-dependent Schrödinger equation, time fractional Schrödinger equation, distributed order time fractional Sch...

In this survey, we review the historical development for the Carleson problem about the a.e. pointwise convergence in five aspects: the a.e. convergence for generalized Schrödinger operators along vertical lines; a.e. convergence for Schröd...

The Schrödinger equation is one of the most important equations in physics and chemistry and can be solved in the simplest cases by computer numerical methods. Since the beginning of the 1970s, the computer began to be used to solve this equatio...

Some types of the generalized nonlinear Schrödinger equation of the second, fourth and sixth order are considered. The Cauchy problem for equations in the general case cannot be solved by the inverse scattering transform. The main objective of t...

The multitemporal nonlinear Schrödinger PDE (with oblique derivative) was stated for the first time in our research group as a universal amplitude equation which can be derived via a multiple scaling analysis in order to describe slow modulations of...

In this paper, we first study the uniqueness and symmetry of solution of nonlinear Schrödinger–Kirchhoff equations with constant coefficients. Then, we show the uniqueness of the solution of nonlinear Schrödinger–Kirchhoff equat...

Schrödinger’s operator relations combined with Einstein’s special relativistic energy-momentum equation produce the linear Klein–Gordon partial differential equation. Here, we extend both the operator relations and the energy-m...

In this paper, a new method for the exact solution of the stationary, one-dimensional Schrödinger equation is proposed. Application of the method leads to a three-parametric family of exact solutions, previously known only in the limiting cases. The...

We propose a new approach that allows one to reduce nonlinear equations on Lie groups to equations with a fewer number of independent variables for finding particular solutions of the nonlinear equations. The main idea is to apply the method of nonco...

In this paper, we consider the time-dependent Schrödinger equation: $i\frac{\partial \psi (x,t)}{\partial t}=\frac{1}{2}{(-\mathrm{\Delta })}^{\frac{\alpha }{2}}\psi (x,t)+V\left(x\right)\psi (x,t),$ $x\in \mathbb{R},t>0$ with the Riesz space-fractional derivative of order...

We re-examine the mathematical properties of the kink and antikink soliton solutions to the Logarithmic Schrödinger Equation (LogSE), a nonlinear logarithmic version of the Schrödinger Equation incorporating Everett–Hirschman entropy....

We study symmetry properties and the possibility of exact integration of the time-independent Schrödinger equation in an external electromagnetic field. We present an algorithm for constructing the first-order symmetry algebra and describe its struct...

The influence of a strong and gusty wind field on ocean waves is investigated. How the random wind affects solitary waves is analyzed in order to obtain insights about wave generation by randomly time varying wind forcing. Using the Euler equations o...

The authors have recently investigated a type of Hyers–Ulam stability of one-dimensional time-independent Schrödinger equation with a symmetric parabolic potential wall. In this paper, we investigate a type of Hyers–Ulam stability of...

The aim of this paper was to propose a systematic study of a $(1+1)$ -dimensional higher order nonlinear Schrödinger equation, arising in two different contexts regarding the biological science and the nonlinear optics. We performed a L...

In many physical contexts, notably including deep-water waves, modulation instability in one space dimension is often studied by using the nonlinear Schrödinger equation. The principal solutions of interest are solitons and breathers which are a...

A type of Hyers–Ulam stability of the one-dimensional, time independent Schrödinger equation was recently investigated; the relevant system had a parabolic potential wall. As a continuation, we proved a type of Hyers–Ulam stability o...

This study considers a class of backward stochastic semi-linear Schrödinger equations with Poisson jumps in ${\mathbb{R}}^d$ or in its bounded domain of a $C^2$ boundary, which is associated with a stochastic control problem of nonlinear Schrödinger equatio...

We construct the Green functions (or Feynman’s propagators) for the Schrödinger equations of the form $i{\psi }_t+{\displaystyle \frac{1}{4}}{\psi }_{xx}\pm t{x}^2\psi =0$ (for the wave function $\psi$ and its time (t) and x-space derivatives) in terms o...

This paper explains how the so-called Einstein locality is to be understood in the Schrödinger picture of quantum mechanics. This notion is fully compatible with the Bell non-locality exhibited by entangled states. Contrary to the belief that qu...

The Schrödinger bridge problem (SBP) finds the most likely stochastic evolution between two probability distributions given a prior stochastic evolution. As well as applications in the natural sciences, problems of this kind have important applicatio...

We consider novel discrete derivative nonlinear Schrödinger equations (ddNLSs). Taking the continuum derivative nonlinear Schrödinger equation (dNLS), we use for the discretisation of the derivative the forward, backward, and central differ...

We review recent results on how to extend the supersymmetry SUSY normalism in Quantum Mechanics to linear generalizations of the time-dependent Schrödinger equation in (1+1) dimensions. The class of equations we consider contains many known cases, su...

Here, we investigate a three-dimensional Schrödinger equation that generalizes the standard framework by incorporating geometric constraints. Specifically, the equation is adapted to account for a backbone structure exhibiting memory effects dependen...

The first author has recently investigated a type of Hyers-Ulam stability of the one-dimensional time independent Schrödinger equation when the relevant system has a rectangular potential barrier of finite height. In the present paper, we will i...

We have considered the Iwasawa and Gauss decompositions for the Lie group SL(2,R). According to these decompositions, the Casimir operators of the group and the Hamiltonians with position-dependent mass were expressed. Then, the unbound state solutio...

Let be the high-order Schrödinger operator ${(-\Delta )}^2+V^2,$ where V is a non-negative potential satisfying the reverse Hölder inequality ($R{H}_q$), with $q>n/2$ and $n\ge 5$. In this paper, we prove that when $0<\alpha \le 2-n/q$, th...

We investigate ground states of a (nonlocal) nonlinear Schrödinger equation which generalizes classical (fractional, relativistic, etc.) Schrödinger equations, so that we extend relevant results and study common properties of these equation...

Many dimensionality and model reduction techniques rely on estimating dominant eigenfunctions of associated dynamical operators from data. Important examples include the Koopman operator and its generator, but also the Schrödinger operator. We p...

In optical receiver mode (ORM) division multiplexing optical communication systems, which can ultimately achieve a very high spectral efficiency, an accurate evaluation of the ORMs is crucial. Conventionally, to find the mode functions and the eigenv...

We study a separable Hilbert space of smooth curves taking values in the Segal–Bergmann space of analytic functions in the complex plane, and two of its subspaces that are the domains of unbounded non self-adjoint linear partial differential op...

Nonlocal nonlinear Schrödinger equations are among the important models of nonlocal integrable systems. This paper aims to present a general formula for arbitrary-order breather solutions to multi-component nonlocal nonlinear Schrödinger eq...

We derive new kinetic and a porous medium equations from the nonlinear Schrödinger equation with random potentials. The kinetic equation has a very similar form compared to the four-wave turbulence kinetic equation in the wave turbulence theory....

In this work, a class of perturbed nonlinear Schrödinger equation is studied by using the homotopy perturbation method. Firstly, we obtain some Jacobi-like elliptic function solutions of the corresponding typical general undisturbed nonlinear Schrödi...

This paper introduces two high-accuracy linearly implicit conservative schemes for solving the nonlocal Schrödinger equation, employing the extrapolation technique. These schemes are based on the generalized scalar auxiliary variable approach an...

The mass and momentum transfer phenomena in a compressible fluid represented by the Navier–Stokes equations are shown to convert into the Schrödinger equation for quantum mechanics. The complete Navier–Stokes equations render into an extended general...

Motivated by the limited work performed on the development of computational techniques for solving the nonlinear Schrödinger equation with time-dependent coefficients, we develop a modified Runge–Kutta pair with improved periodicity and st...

We derive the functional Schrödinger equation for quantum fields in curved spacetime in the semiclassical limit of quantum geometrodynamics with a Gaussian incoherent dust acting as a clock field. We perform the semiclassical limit using a WKB-t...

A numerical method for evolving the nonlinear Schrödinger equation on a coarse spatial grid is developed. This trains a neural network to generate the optimal stencil weights to discretize the second derivative of solutions to the nonlinear Schr...

In this paper, we propose two efficient methods for solving the fractional-order Schrödinger–KdV system. The first method is the Laplace residual power series method (LRPSM), which involves expressing the solution as a power series and usi...

We investigate the effect of giant negative magnetoresistance in ultrahigh-mobility ($\mu \gg {10}^7{\mathrm{cm}}^2{\mathrm{V}}^{-1}{\mathrm{s}}^{-1}$) two-dimensional electron systems. These systems present a dramatic drop in the mangetoresistance at low magnetic fields ($B\sim 0$...

A control problem with final overdetermination is considered for the higher-order nonlinear Schrödinger equation on a bounded interval. The boundary condition on the space derivative is chosen as the control. Results on the global existence of s...

The Lie symmetry analysis for the study of a $1+n$ fourth-order Schrödinger equation inspired by the modification of the deformation algebra in the presence of a minimum length is applied. Specifically, we perform a detailed classification for the...