Transactions in Theoretical and Mathematical Physics

Transactions in Theoretical and Mathematical Physics

Balance Laws and Soliton Persistence in a Nonlinear Schrödinger Equation with Delayed Kerr Response

Document Type : Original Article

Authors
Anouchah Latifi 1
Erfane Namvar 2
1 Qom University of Technology
2 Department of Mechanics, Qom University of Technology, Qom, Iran
10.30511/ttmp.2026.2087308.1073
Abstract
We study the persistence and slow evolution of soliton
solutions of a nonlinear Schrödinger equation perturbed by a
nonlinear intensity–gradient term arising from a weakly
noninstantaneous Kerr response. Starting from Maxwell’s
equations with a nonlocal nonlinear polarization, we derive a
perturbed envelope equation containing the correction
\( i\eta\,\psi\,\partial_t |\psi|^2 \), which represents the
leading contribution of a short-memory nonlinear response.
To analyze the resulting dynamics we employ a variational
collective–coordinate reduction that describes the pulse in
terms of a small set of evolving soliton parameters. The
reduced dynamical system shows that the perturbation preserves
the optical power while producing a slow evolution of the
soliton center and carrier frequency. At the level of the
governing partial differential equation we derive an exact
balance law for the momentum, which reveals that the nonlinear
gradient term acts as a systematic source of momentum drift.
This balance relation yields explicit scaling predictions for
the long-distance evolution of the soliton parameters.
Numerical simulations confirm the persistence of a localized
pulse together with the predicted parameter drift. The results
provide a transparent connection between the microscopic origin
of delayed nonlinear responses, the modified conservation
structure of the perturbed equation, and the observable
dynamics of optical solitons.
Keywords
nonlinear Schrodinger equation
optical solitons
delayed Kerr response
balance laws
collective coordinates
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Transactions in Theoretical and Mathematical Physics
Volume 3, Issue 1
Winter 2026
Pages 34-41

  • Receive Date 22 January 2026
  • Accept Date 08 February 2026
  • First Publish Date 08 February 2026
  • Publish Date 01 February 2026