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Updated: Jun 18, 2026

Generation and Coherent Control of Pulsed Quantum Frequency Combs
Published on: June 8, 2018
Exact solutions to three-dimensional generalized nonlinear Schrödinger equations with varying potential and
1Centro de Física Teórica e Computacional, Universidade de Lisboa, Complexo Interdisciplinar, Avenida Professor Gama Pinto 2, Lisboa 1649-003, Portugal. zhenyan@cii.fc.ul.pt
Researchers reduced complex 3D nonlinear Schrödinger (NLS) equations to a simpler 1D form using similarity transformations. This method enables finding exact localized and periodic wave solutions for NLS equations with varying coefficients.
Area of Science:
- Mathematical Physics
- Nonlinear Dynamics
- Wave Phenomena
Background:
- Nonlinear Schrödinger (NLS) equations model various wave phenomena.
- Solving inhomogeneous NLS equations in higher dimensions is challenging.
Purpose of the Study:
- To develop a method for reducing 3D inhomogeneous NLS equations to a 1D stationary form.
- To obtain exact localized and periodic wave solutions.
Main Methods:
- Application of similarity transformations.
- Mapping 3D coordinates to one-parametric coordinate surfaces.
- Analysis of power-law potentials (linear and nonlinear).
Main Results:
- Successfully reduced 3D inhomogeneous p-q NLS equations to a 1D stationary NLS equation.
- Demonstrated the ability to find exact localized and periodic wave solutions.
- Presented case examples for power-law potentials.
Conclusions:
- The similarity transformation method provides an effective approach to solve complex NLS equations.
- The technique allows for the derivation of exact solutions for a range of potential types.
- Generalizations of the method are feasible for broader applications.
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