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A Hyperbolic Approximation of the Nonlinear Schrödinger Equation

Abhijit Biswas, Laila S. Busaleh, David I. Ketcheson, Carlos Muñoz-Moncayo, Manvendra Rajvanshi

TL;DR

This work introduces NLSH, a first-order hyperbolic relaxation of the nonlinear Schrödinger equation, and proves its strict hyperbolicity alongside a modified Hamiltonian framework with at least three conserved quantities that approximate the NLS invariants. It derives explicit standing-wave solutions for NLSH that converge to NLS ground states as the relaxation parameter $\tau\to0$, and develops asymptotic-preserving, mass-conserving discretizations based on ImEx Runge-Kutta schemes that remain stable and consistent in the stiff limit while recovering a discretization of NLS. Numerical experiments in both focusing and defocusing regimes validate the AP/AA properties, demonstrate accurate soliton and front approximations, and show the practical benefits of relaxation-enhanced conservation for long-time integration. Overall, the approach preserves more of the original Hamiltonian structure than prior hyperbolic approximations and suggests promising extensions to higher dimensions and nonperiodic boundaries, with open questions about additional invariants and modulation phenomena.

Abstract

We study a first-order hyperbolic approximation of the nonlinear Schrödinger (NLS) equation. We show that the system is strictly hyperbolic and possesses a modified Hamiltonian structure, along with at least three conserved quantities that approximate those of NLS. We provide families of explicit standing-wave solutions to the hyperbolic system, which are shown to converge uniformly to ground-state solutions of NLS in the relaxation limit. The system is formally equivalent to NLS in the relaxation limit, and we develop asymptotic preserving discretizations that tend to a consistent discretization of NLS in that limit, while also conserving mass. Examples for both the focusing and defocusing regimes demonstrate that the numerical discretization provides an accurate approximation of the NLS solution.

A Hyperbolic Approximation of the Nonlinear Schrödinger Equation

TL;DR

This work introduces NLSH, a first-order hyperbolic relaxation of the nonlinear Schrödinger equation, and proves its strict hyperbolicity alongside a modified Hamiltonian framework with at least three conserved quantities that approximate the NLS invariants. It derives explicit standing-wave solutions for NLSH that converge to NLS ground states as the relaxation parameter , and develops asymptotic-preserving, mass-conserving discretizations based on ImEx Runge-Kutta schemes that remain stable and consistent in the stiff limit while recovering a discretization of NLS. Numerical experiments in both focusing and defocusing regimes validate the AP/AA properties, demonstrate accurate soliton and front approximations, and show the practical benefits of relaxation-enhanced conservation for long-time integration. Overall, the approach preserves more of the original Hamiltonian structure than prior hyperbolic approximations and suggests promising extensions to higher dimensions and nonperiodic boundaries, with open questions about additional invariants and modulation phenomena.

Abstract

We study a first-order hyperbolic approximation of the nonlinear Schrödinger (NLS) equation. We show that the system is strictly hyperbolic and possesses a modified Hamiltonian structure, along with at least three conserved quantities that approximate those of NLS. We provide families of explicit standing-wave solutions to the hyperbolic system, which are shown to converge uniformly to ground-state solutions of NLS in the relaxation limit. The system is formally equivalent to NLS in the relaxation limit, and we develop asymptotic preserving discretizations that tend to a consistent discretization of NLS in that limit, while also conserving mass. Examples for both the focusing and defocusing regimes demonstrate that the numerical discretization provides an accurate approximation of the NLS solution.

Paper Structure

This paper contains 20 sections, 6 theorems, 91 equations, 8 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Hamiltonian structure
  3. NLS as a Hamiltonian system
  4. NLSH as a Hamiltonian system
  5. Standing-wave solutions
  6. Standing fronts
  7. Standing solitary waves
  8. Asymptotic-Preserving and Energy-Conserving Numerical Discretization
  9. Mass-Conserving Spatial Semidiscretization
  10. ImEx Runge-Kutta Time Discretization
  11. Asymptotic-preserving time discretization
  12. Fully-discrete mass conservation via temporal relaxation
  13. Numerical tests
  14. Verification of Asymptotic Preserving Property
  15. AP results for type I ImEx-RK methods
  16. ...and 5 more sections

Key Result

Theorem 1

The NLSH equations nlsH with $\tau>0$ are strictly hyperbolic and possess three conserved quantities: $\bar{H}, \bar{I}_1, \bar{I}_2$.

Figures (8)

  • Figure 1: Phase portraits illustrating the ground state solutions of the NLSH system with $\tau=10^{-3}$ and $\mu=\kappa$
  • Figure 2: Standing front solutions to the NLSH system \ref{['eq:Exact_traveling_front_NLSH']} and the NLS equation \ref{['eq:Exact_traveling_front_NLS']} with $\kappa=\mu=-1$, $K_0=0$, and different values of $\tau$
  • Figure 3: Solitary wave solutions to the NLSH system \ref{['eq:explicit_solitary_NLSH_q0']} and the NLS equation \ref{['eq:explicit_solitary_NLS']} with $\kappa=\mu=1$, $K_1=0$, and different values of $\tau$
  • Figure 4: A schematic illustration of the AP property biswas2024kdvh.
  • Figure 5: Bound state soliton solutions for 2 (left) and 3 (right) solitons. As $\tau\to 0$ the NLSH solution converges to NLS solution.
  • ...and 3 more figures

Theorems & Definitions (15)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Proposition 1
  • proof
  • Proposition 2
  • Theorem 2
  • proof
  • Definition 1
  • Definition 2
  • ...and 5 more