Stability of Solitons in the Parametrically Driven and Damped Nonlinear Schrödinger Equation in Arbitrary Dimension
Abstract
We study the spectral stability of soliton solutions of the parametrically driven and damped nonlinear Schrödinger (PDDNLS) equation with power-law nonlinearity |Ψ|2κΨ in N dimensions. We identify two families of stationary solutions, Ψ±, whose existence is determined by two conditions: one involving the nonlinearity exponent~κ and the spatial dimension~N, and another involving the forcing amplitude~r and the damping coefficient~ρ. By linearising around these solutions and introducing a convenient parametrisation through~ε, we reduce the stability problem to a dissipationless eigenvalue problem for two Schrödinger-type operators, L0 and L1. We prove that Ψ- is always unstable, while for Ψ+, we carry out a complete spectral analysis by decomposing the problem into angular momentum sectors 𝒮l. We show that modes with l ≥ 2 generate no instabilities and that no complex quadruplets arise on~𝒮1. We also derive an analytical instability criterion on 𝒮l by combining a Vakhitov--Kolokolov-type argument with a local analysis that determines the direction of the eigenvalue splitting near the origin. In addition, we uncover new phenomena at both small and large values of κ that were not investigated in earlier one-dimensional studies. Numerical stability diagrams for N = 1, 2, 3 are presented.
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BibTeX
@misc{carrenonavas2026stability,
title = {{Stability of Solitons in the Parametrically Driven and Damped Nonlinear Schrödinger Equation in Arbitrary Dimension}},
author = {Fernando Carreño-Navas},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00346},
eprint = {2611.00346},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00346},
}
RIS
TY - UNPB
TI - Stability of Solitons in the Parametrically Driven and Damped Nonlinear Schrödinger Equation in Arbitrary Dimension
AU - Fernando Carreño-Navas
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00346
UR - https://arxiv.org/abs/2611.00346
AB - We study the spectral stability of soliton solutions of the parametrically driven and damped nonlinear Schrödinger (PDDNLS) equation with power-law nonlinearity |Ψ|^2κΨ in N dimensions. We identify two families of stationary solutions, Ψ_±, whose existence is determined by two conditions: one involving the nonlinearity exponent~κ and the spatial dimension~N, and another involving the forcing amplitude~r and the damping coefficient~ρ. By linearising around these solutions and introducing a convenient parametrisation through~ε, we reduce the stability problem to a dissipationless eigenvalue problem for two Schrödinger-type operators, L_0 and L_1. We prove that Ψ_- is always unstable, while for Ψ_+, we carry out a complete spectral analysis by decomposing the problem into angular momentum sectors 𝒮_l. We show that modes with l ≥ 2 generate no instabilities and that no complex quadruplets arise on~𝒮_1. We also derive an analytical instability criterion on 𝒮_l by combining a Vakhitov--Kolokolov-type argument with a local analysis that determines the direction of the eigenvalue splitting near the origin. In addition, we uncover new phenomena at both small and large values of κ that were not investigated in earlier one-dimensional studies. Numerical stability diagrams for N = 1, 2, 3 are presented.
ER -