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Exact solutions of the inhomogeneous nonlinear Schrödinger equation through supersymmetric potentials

David J. Fernández C., O. Pavón-Torres

TL;DR

The authors address finding exact stationary solutions to the INLSE with spatially varying nonlinearity and external potentials by combining Lie symmetry analysis with supersymmetric quantum mechanics. By mapping the INLSE to a standard NLSE via a canonical transformation, they transform the problem into a solvable form and then use SUSY QM to generate partner potentials from a simple starting point, notably obtaining the Pösch-Teller potential with a single bound state. The approach yields explicit inhomogeneous nonlinearities and corresponding soliton solutions, including elliptic-function and kink-type profiles, under various SUSY scenarios. This framework enables systematic construction of exact INLSE solutions and can be extended to higher-order transformations and complex potentials, with potential applications in Bose-Einstein condensates and nonlinear optics where spatial modulation and PT-symmetric effects are relevant.

Abstract

By employing supersymmetric quantum mechanics, we present a general algorithm to construct supersymmetric partner potentials and hence derive exact stationary solutions of the inhomogeneous nonlinear Schrödinger equation (INLSE). This is possible due to the connection between the INLSE and the nonlinear Schrödinger equation (NLSE), which can be established from a treatment based on Lie point symmetries and is related with Schrödinger equation, under certain conditions. As an illustrative example, we construct exact solutions for the INLSE through a Pösch-Teller potential with a single bound state.

Exact solutions of the inhomogeneous nonlinear Schrödinger equation through supersymmetric potentials

TL;DR

The authors address finding exact stationary solutions to the INLSE with spatially varying nonlinearity and external potentials by combining Lie symmetry analysis with supersymmetric quantum mechanics. By mapping the INLSE to a standard NLSE via a canonical transformation, they transform the problem into a solvable form and then use SUSY QM to generate partner potentials from a simple starting point, notably obtaining the Pösch-Teller potential with a single bound state. The approach yields explicit inhomogeneous nonlinearities and corresponding soliton solutions, including elliptic-function and kink-type profiles, under various SUSY scenarios. This framework enables systematic construction of exact INLSE solutions and can be extended to higher-order transformations and complex potentials, with potential applications in Bose-Einstein condensates and nonlinear optics where spatial modulation and PT-symmetric effects are relevant.

Abstract

By employing supersymmetric quantum mechanics, we present a general algorithm to construct supersymmetric partner potentials and hence derive exact stationary solutions of the inhomogeneous nonlinear Schrödinger equation (INLSE). This is possible due to the connection between the INLSE and the nonlinear Schrödinger equation (NLSE), which can be established from a treatment based on Lie point symmetries and is related with Schrödinger equation, under certain conditions. As an illustrative example, we construct exact solutions for the INLSE through a Pösch-Teller potential with a single bound state.

Paper Structure

This paper contains 8 sections, 82 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Lie symmetry analysis for the INLSE
  3. INLSE with supersymmetric potentials
  4. Supersymmetric Quantum Mechanics
  5. An exactly solvable example
  6. Case with $\lambda_{1}=\lambda_{0}$
  7. Case with $\lambda_{1}\neq\lambda_{0}$
  8. Conclusion

Figures (4)

  • Figure 1: (a) Spatially inhomogeneous nonlinearity $g(x)$. (b) Soliton profile for the Pösch-Teller potential with one bound state. The chosen parameters are $k_{0}=0.5$, $\alpha=2$, $\gamma=3$ and $\beta=g_{0}=1$.
  • Figure 2: (a) Spatially inhomogeneous nonlinearity $g(x)$. (b) Dark soliton profile for the Pösch-Teller potential with one bound state. The chosen parameters are $k_{0}=0.5$, $\alpha=2$, $\gamma=3$ and $\beta=g_{0}=1$.
  • Figure 3: (a) Spatially inhomogeneous nonlinearity $g(x)$. (b) Dark soliton profile for the Pösch-Teller potential with one bound state. The chosen parameters are $k_{0}=0.5$, $k_{1}=0.4$, $\alpha=2$, $\gamma=3$ and $\beta=g_{0}=1$.
  • Figure 4: (a) Spatially inhomogeneous nonlinearity $g(x)$. (b) Dark soliton profile for the Pösch-Teller potential with one bound state. The chosen parameters are $k_{0}=0.5$, $k_{1}=0.4$, $\alpha=2$, $\gamma=3$ and $\beta=g_{0}=1$.