Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Modern Physics Letters B

Conditions for a Solitonic Solution of the Doener–Goldin Equation of Quantum Mechanics

Elisabeth C. Caparelli, Salomon Sylvain Mizrahi, V. V. Dodonov

Published Jun 10, 1998• 2 citations
Publisher

Abstract

A solitonic solution to the free-particle motion of Doebner–Goldin nonlinear equation is shown to exist under special conditions. For small values of the nonlinearity parameters, the solution is a plane wave modulated by a cos function, and the solitonic one arises when the parameters surpass some critical values. The mean energy of the particle is a conserved quantity and the continuity equation holds. We also verify that there is no nonlinear gauge transformation (in the sense of Ref. 12) that linearizes that equation.

DOI
10.1142/s0217984998000627

Related papers

Nonlinear Gauge Transformations and Exact Solutions of the Doebner-Goldin Equation

P. Nattermann, Wolfgang Scherer

Invariants of nonlinear gauge transformations of a family of nonlinear Schrödinger equations proposed by Doebner and Goldin are used to characterize the behaviour of exact solutions of these equations.

arXiv:quant-ph/9506033•Jun 21, 1995•8 citations

Quantum particle model of free particle solution in one-dimensional Klein-Gordon equation

Teguh Budi Prayitno

We have discussed the quantum particle model for the case of a free particle solution in the one-dimensional Klein-Gordon nonlinear equations. This model was obtained through the two equations from the conservation laws of the classical physics, namely, the Hamilton-Jacobi equation for the relativistic motion and continuity equations. In this case, the Hamilton-Jacobi equation describes a part of the particle while the continuity equation describes the wave side. The derivation of this equation did not use the two postulates in the quantum mechanics, namely, the Einstein and de Broglie’s postulates regarding the quantization of energy and momentum. According to this derivation, the particle side has almost most of the energy of quantum particle that accumulate at a point while the wave part has only a small portion of the energy of the quantum particle that surrounds the part of the particle. In addition, this paper also shows the form of mathematical functions that represent the particle and wave parts for a free particle solution. This form is obtained through a special solution of the free particle which is a plane-wave solution.

AIP conference proceedings•Jan 1, 2019•2 citations

Soliton-like solutions to the ordinary Schrödinger equation within standard quantum mechanics

Michel Zamboni‐Rached, Erasmo Recami

In recent times attention has been paid to the fact that (linear) wave equations admit of “soliton-like” solutions, known as localized waves or non-diffracting waves, which propagate without distortion in one direction. Such localized solutions (existing also for K-G or Dirac equations) are a priori suitable, more than gaussian's, for describing elementary particle motion. In this paper we show that, mutatis mutandis, localized solutions exist even for the ordinary (linear) Schrödinger equation within standard quantum mechanics; and we obtain both approximate and exact solutions, also setting forth for them particular examples. In the ideal case such solutions (even if localized and “decaying”) are not square-integrable, as well as plane or spherical waves: we show therefore how to obtain finite-energy solutions. At last, we briefly consider solutions for a particle moving in the presence of a potential.

arXiv:1008.3087•Journal of Mathematical Physics•May 1, 2012•10 citations

On Solutions to the Nonlinear Phase Modification of the Schroedinger Equation

Waldemar Puszkarz

We present some physically interesting, in general non-stationary, one-dimensional solutions to the nonlinear phase modification of the Schrödinger equation proposed recently. The solutions include a coherent state, a phase-modified Gaussian wave packet in the potential of harmonic oscillator whose strength varies in time, a free Gaussian soliton, and a similar soliton in the potential of harmonic oscillator comoving with the soliton. The last of these solutions implies that there exist an energy level in the spectrum of harmonic oscillator which is not predicted by the linear theory. The free solitonic solution can be considered a model for a particle aspect of the wave-particle duality embodied in the quantum theory. The physical size of this particle is naturally rendered equal to its Compton wavelength in the subrelativistic framework in which the self-energy of the soliton is assumed to be equal to its rest-mass energy. The solitonic solutions exist only for the negative coupling constant for which the Gaussian wave packets must be larger than some critical finite size if their energy is to be bounded, i.e., they cannot be point-like objects.

arXiv:quant-ph/9903010•Mar 4, 1999•1 citation

The Free Particle Solution for NonLinear Master Schrödinger

Teguh Budi Prayitno

The solution of free particle has been studied by solving nonlinear master Schrodinger equation. This equation actually consists of two properties, Hamilton-Jacobi equation and continuity equation which describe two classical balance equations. In other words, those two equations were assumed to be hold in quantum level. Unfortunately, there are several properties in ordinary quantum mechanics must be ignored, such as first principle quantization and superposition principle. In this paper we briefly review some important study of the characteristics of nonlinear master Schrodinger equation. In addition, we also have proposed two kinds of free particle solution for the nonlinear master Schrodinger and founded that the solution poses a plane wave solution. Keywords: nonlinear master Schrodinger, free particle, Hamilton-Jacobi.

•Dec 27, 2010•0 citations

Higher Order Modification of the Schroedinger Equation

Waldemar Puszkarz

We modify the Schrödinger equation in a way that preserves its main properties but makes use of higher order derivative terms. Although the modification represents an analogy to the Doebner-Goldin modification, it can differ from it quite distinctively. A particular model of this modification including derivatives up to the fourth order is examined in greater detail. We observe that a special variant of this model partially retains the linear superposition principle for the wave packets of standard quantum mechanics remain solutions to it. It is a peculiarity of this variant that a periodic structure emerges naturally from its equations. As a result, a free particle, in addition to a plane wave solution, can possess band solutions. It is argued that this can give rise to well-focused particle trajectories. Owing to this peculiarity, when interpreted outside quantum theory, the equations of this modification could also be used to model pattern formation phenomena.

arXiv:quant-ph/9710007•Oct 2, 1997•1 citation
A solitonic solution to the free-particle motion of Doebner–Goldin nonlinear equation is shown to exist under special conditions. For small values of the nonlinearity parameters, the solution is a plane wave modulated by a cos function, and the solitonic one arises when the parameters surpass some critical values. The mean energy of the particle is a conserved quantity and the continuity equation holds. We also verify that there is no nonlinear gauge transformation (in the sense of Ref. 12) that linearizes that equation.
We have discussed the quantum particle model for the case of a free particle solution in the one-dimensional Klein-Gordon nonlinear equations. This model was obtained through the two equations from the conservation laws of the classical physics, namely, the Hamilton-Jacobi equation for the relativistic motion and continuity equations. In this case, the Hamilton-Jacobi equation describes a part of the particle while the continuity equation describes the wave side. The derivation of this equation did not use the two postulates in the quantum mechanics, namely, the Einstein and de Broglie’s postulates regarding the quantization of energy and momentum. According to this derivation, the particle side has almost most of the energy of quantum particle that accumulate at a point while the wave part has only a small portion of the energy of the quantum particle that surrounds the part of the particle. In addition, this paper also shows the form of mathematical functions that represent the particle and wave parts for a free particle solution. This form is obtained through a special solution of the free particle which is a plane-wave solution.
In recent times attention has been paid to the fact that (linear) wave equations admit of “soliton-like” solutions, known as localized waves or non-diffracting waves, which propagate without distortion in one direction. Such localized solutions (existing also for K-G or Dirac equations) are a priori suitable, more than gaussian's, for describing elementary particle motion. In this paper we show that, mutatis mutandis, localized solutions exist even for the ordinary (linear) Schrödinger equation within standard quantum mechanics; and we obtain both approximate and exact solutions, also setting forth for them particular examples. In the ideal case such solutions (even if localized and “decaying”) are not square-integrable, as well as plane or spherical waves: we show therefore how to obtain finite-energy solutions. At last, we briefly consider solutions for a particle moving in the presence of a potential.
We present some physically interesting, in general non-stationary, one-dimensional solutions to the nonlinear phase modification of the Schrödinger equation proposed recently. The solutions include a coherent state, a phase-modified Gaussian wave packet in the potential of harmonic oscillator whose strength varies in time, a free Gaussian soliton, and a similar soliton in the potential of harmonic oscillator comoving with the soliton. The last of these solutions implies that there exist an energy level in the spectrum of harmonic oscillator which is not predicted by the linear theory. The free solitonic solution can be considered a model for a particle aspect of the wave-particle duality embodied in the quantum theory. The physical size of this particle is naturally rendered equal to its Compton wavelength in the subrelativistic framework in which the self-energy of the soliton is assumed to be equal to its rest-mass energy. The solitonic solutions exist only for the negative coupling constant for which the Gaussian wave packets must be larger than some critical finite size if their energy is to be bounded, i.e., they cannot be point-like objects.
The solution of free particle has been studied by solving nonlinear master Schrodinger equation. This equation actually consists of two properties, Hamilton-Jacobi equation and continuity equation which describe two classical balance equations. In other words, those two equations were assumed to be hold in quantum level. Unfortunately, there are several properties in ordinary quantum mechanics must be ignored, such as first principle quantization and superposition principle. In this paper we briefly review some important study of the characteristics of nonlinear master Schrodinger equation. In addition, we also have proposed two kinds of free particle solution for the nonlinear master Schrodinger and founded that the solution poses a plane wave solution. Keywords: nonlinear master Schrodinger, free particle, Hamilton-Jacobi.
We modify the Schrödinger equation in a way that preserves its main properties but makes use of higher order derivative terms. Although the modification represents an analogy to the Doebner-Goldin modification, it can differ from it quite distinctively. A particular model of this modification including derivatives up to the fourth order is examined in greater detail. We observe that a special variant of this model partially retains the linear superposition principle for the wave packets of standard quantum mechanics remain solutions to it. It is a peculiarity of this variant that a periodic structure emerges naturally from its equations. As a result, a free particle, in addition to a plane wave solution, can possess band solutions. It is argued that this can give rise to well-focused particle trajectories. Owing to this peculiarity, when interpreted outside quantum theory, the equations of this modification could also be used to model pattern formation phenomena.