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 $q$.

💡 Research Summary

The authors introduce a novel nonlinear Schrödinger framework based on a q‑dependent nonlinear derivative operator D₍q₎. This operator, defined as D₍q₎f(x)= (1/q) f(x)^{q‑1} d f(x)^{q‑1}/dx, reduces to the ordinary derivative when the deformation parameter q approaches unity. By extending this operator to spatial gradients (∇₍q₎) and a temporal derivative (D₍q₎ₜ), they construct a Lagrangian density
L = i ℏ Ψ*^{q} D₍q₎ₜ Ψ – (ℏ²/2m) ∇Ψ*·∇Ψ – V( x ) |Ψ|²,
where Ψ is a normalized wavefunction and the q‑powers introduce nonlinearity directly into the kinetic sector rather than through a potential term. Applying the Euler‑Lagrange equations yields the q‑deformed nonlinear Schrödinger equation (q‑NLSE):
i ℏ ∂ₜΨ + (ℏ²/2m) Ψ*^{1‑q} ∇²Ψ – V Ψ*^{1‑q} Ψ = 0.
Thus, the kinetic term is multiplied by a factor Ψ*^{1‑q}, which collapses to the standard linear term when q→1.

Gauge invariance is examined by coupling the field to an electromagnetic four‑potential (A, A₀). Covariant derivatives are defined as
D = ∇ – i e A, Dₜ = ∂ₜ + i e q A₀,
so that under a local U(1) transformation Ψ→e^{i e θ(x,t)}Ψ the Lagrangian remains invariant provided the potentials transform in the usual way. The presence of q in the temporal covariant derivative effectively rescales the electric charge to q e, implying that physically realistic scenarios require q to be close to unity.

The probability density is redefined as ρ = |Ψ|^{2q}, and the associated current takes the familiar form j = (ℏ/2mi)