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arXiv:2602.11312

A Nonlinear $q$-Deformed Schrödinger Equation

M. A. Rego-Monteiro, E. M. F. Curado

nlin.PSmath-phquant-phSubmitted Feb 11, 2026• Announced Feb 13, 2026

Abstract

We construct a new nonlinear deformed Schrödinger structure using a nonlinear derivative operator which depends on a parameter $q$. This operator recovers Newton derivative when $q \rightarrow 1$. Using this operator we propose a deformed Lagrangian which gives us a deformed nonlinear Schrödinger equation with a nonlinear kinetic energy term and a standard potential $V(\vec{x})$. We analytically solve the nonlinear deformed Schrödinger equation for $V(\vec{x}) = 0$ and $q \simeq1$. This model has a continuity equation, the energy is conserved, as well as the momentum and also interacts with electromagnetic field. Planck relation remains valid and in all steps we easily recover the undeformed quantities when the deformation parameter goes to 1. Finally, we numerically solve the equation of motion for the free particle in any spatial dimension, which shows a solitonic pattern when the space is equal to one for particular values of $q$.

Journal reference: Physics Letters A 565 (2026) 131100
DOI: 10.1016/j.physleta.2025.131100
Comments: 20 pages, 8 figures

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A Nonlinear

q

-Deformed Schrödinger Equation

We construct a new nonlinear deformed Schrödinger structure using a nonlinear derivative operator which depends on a parameter

q

. This operator recovers Newton derivative when

q \rightarrow 1

. Using this operator we propose a deformed Lagrangian which gives us a deformed nonlinear Schrödinger equation with a nonlinear kinetic energy term and a standard potential

V(\vec{x})

. We analytically solve the nonlinear deformed Schrödinger equation for

V(\vec{x}) = 0

and

q \simeq1

. This model has a continuity equation, the energy is conserved, as well as the momentum and also interacts with electromagnetic field. Planck relation remains valid and in all steps we easily recover the undeformed quantities when the deformation parameter goes to 1. Finally, we numerically solve the equation of motion for the free particle in any spatial dimension, which shows a solitonic pattern when the space is equal to one for particular values of

q

.

This is the second part of a paper about a q-deformed analog of non-relativistic Schroedinger theory. It applies the general ideas of part I and tries to give a description of one-particle states on q-deformed quantum spaces like the braided line or the q-deformed Euclidean space in three dimensions. Hamiltonian operators for the free q-deformed particle in one as well as three dimensions are introduced. Plane waves as solutions to the corresponding Schroedinger equations are considered. Their completeness and orthonormality relations are written down. Expectation values of position and momentum observables are taken with respect to one-particle states and their time-dependence is discussed. A potential is added to the free-particle Hamiltonians and q-analogs of the Ehrenfest theorem are derived from the Heisenberg equations of motion. The conservation of probability is proved.

An exact classical field theory, for a recently proposed nonlinear generalization of the Schrödinger equation, is presented. In this generalization, a nonlinearity depending on an index q appears in the kinetic term, such that the free-particle linear Schrödinger equation is recovered in the limit q →1. It is shown that besides the usual , a new field must be introduced, which becomes only when q →1. In analogy to the linear case, is interpreted as the probability density for finding the particle at time t , in a given position inside an arbitrary finite volume V , for any q . The possible physical consequences are discussed, and, in particular, the important property that the fields and do not interact with light.

Conservation laws for a

q

-deformed nonrelativistic particle

We derive

q

-versions of Green's theorem from the Leibniz rules of partial derivatives for the

q

-deformed Euclidean space. Using these results and the Schrödinger equations for a

q

-deformed nonrelativistic particle, we derive continuity equations for the probability density, the energy density, and the momentum density of a

q

-deformed nonrelativistic particle.

In the one-dimensional stationary case, we construct a mechanical Lagrangian describing the quantum motion of a nonrelativistic spinless system. This Lagrangian is written as a difference between a function T, which represents the quantum generalization of the kinetic energy and which depends on the coordinate x and the temporal derivatives of x up the third order, and the classical potential V(x). The Hamiltonian is then constructed and the corresponding canonical equations are deduced. The function T is first assumed to be arbitrary. The development of T in a power series together with the dimensional analysis allow us to fix univocally the series coefficients by requiring that the well-known quantum stationary Hamilton–Jacobi equation be reproduced. As a consequence of this approach, we formulate the law of the quantum motion representing a new version of the quantum Newton law. We also analytically establish the famous Bohm relation [Formula: see text] outside the framework of the hydrodynamical approach and show that the well-known quantum potential, although it is a part of the kinetic term, plays really the role of an additional potential as assumed by Bohm.

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 well-localized solitons for free macro-objects but lacks the mechanism of how extended wave functions collapse on solitons. The gravity-related stochastic Schrödinger equation (1989) provides the spontaneous collapse, but the resulting solitons undergo a tiny diffusion, leading to an inconvenient steady increase in the kinetic energy. We propose the stochastic Schrödinger–Newton equation, which contains the above two gravity-related modifications together. Then, the wave functions of free macroscopic bodies will gradually and stochastically collapse to solitons, which perform inertial motion without momentum diffusion: conservation of momentum and energy is restored.

We found hermitian realizations of the position vector

\vec{r}

, angular momentum

\vecΛ

and linear momentum

\vec{p}

behaving like vectors with respect to the

SU_q(2)

algebra, generated by

L_0

and

L_\pm

. They are used to write the

q

deformed Schrödinger equation, whose solution for Coulomb and oscillator potential are briefly discussed.