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The instabilities beyond modulational type in a repulsive Bose-Einstein condensate with a periodic potential
Pattern Formation and Solitons · Submitted 29 Apr 2023
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Based on the defocusing nonlinear Schr\"{o}dinger (NLS) equation with an elliptic function potential, the well-known modulational instability (MI), the more recently identified high-frequency instability, and an unprecedented -- to our knowledge -- variant of the MI, the so-called isola instability are identified numerically.
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The instabilities of the nontrivial phase elliptic solutions in a repulsive Bose-Einstein condensate (BEC) with a periodic potential are investigated. Based on the defocusing nonlinear Schr\"{o}dinger (NLS) equation with an elliptic function potential, the well-known modulational instability (MI), the more recently identified high-frequency instability, and an unprecedented -- to our knowledge -- variant of the MI, the so-called isola instability are identified numerically. Upon varying parameters of the solutions, instability transitions occur through the suitable bifurcations, such as the Hamiltonian Hope one. Specifically, (i) increasing the elliptic modulus $k$ of the solutions, we find that MI switches to the isola instability and the dominant disturbance has twice the elliptic wave's period, corresponding to a Floquet exponent $\mu=\frac{\pi}{2K(k)}$. The isola instability arises from the collision of spectral elements at the origin of the spectral plane. (ii) Upon varying $V_{0}$, the transition between MI and the high-frequency instability occurs. Differently from the MI and isola instability where the collisions of eigenvalues happen at the origin, high-frequency instability arises from pairwise collisions of nonzero, imaginary elements of the stability spectrum; (iii) In the limit of sinusoidal potential, we show that MI occurs from a collision of eigenvalues with $\mu=\frac{\pi}{2K(k)}$ at the origin; (iv) we also examine the dynamic byproducts of the instability in chaotic fields generated by its manifestation. An interesting observation is that, in addition to MI, the isola instability could also lead to dark localized events in the scalar defocusing NLS equation.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The instabilities of the nontrivial phase elliptic solutions in a repulsive Bose-Einstein condensate (BEC) with a periodic potential are investigated. | contribution 0.87 | 0.88 | 0.06 | asserted 0.87 |
| s1 | Based on the defocusing nonlinear Schr\"{o}dinger (NLS) equation with an elliptic function potential, the well-known modulational instability (MI), the more recently identified high-frequency instability, and an unprecedented -- to our knowledge -- variant of the MI, the so-called isola instability are identified numerically. | result 0.53 | 0.89 | 0.36 | asserted 1.00 |
| s2 | Upon varying parameters of the solutions, instability transitions occur through the suitable bifurcations, such as the Hamiltonian Hope one. | result 0.93 | 0.75 | 0.08 | asserted 0.99 |
| s3 | Specifically, (i) increasing the elliptic modulus $k$ of the solutions, we find that MI switches to the isola instability and the dominant disturbance has twice the elliptic wave's period, corresponding to a Floquet exponent $\mu=\frac{\pi}{2K(k)}$. | result 1.00 | 0.96 | 0.21 | asserted 1.00 |
| s4 | The isola instability arises from the collision of spectral elements at the origin of the spectral plane. | result 0.75 | 0.82 | 0.11 | asserted 1.00 |
| s5 | (ii) Upon varying $V_{0}$, the transition between MI and the high-frequency instability occurs. | result 1.00 | 0.88 | 0.07 | asserted 1.00 |
| s6 | Differently from the MI and isola instability where the collisions of eigenvalues happen at the origin, high-frequency instability arises from pairwise collisions of nonzero, imaginary elements of the stability spectrum; (iii) In the limit of sinusoidal potential, we show that MI occurs from a collision of eigenvalues with $\mu=\frac{\pi}{2K(k)}$ at the origin; (iv) we also examine the dynamic byproducts of the instability in chaotic fields generated by its manifestation. | result 0.98 | 0.92 | 0.15 | asserted 1.00 |
| s7 | An interesting observation is that, in addition to MI, the isola instability could also lead to dark localized events in the scalar defocusing NLS equation. | result 0.87 | 0.80 | 0.18 | hedged possibility 0.98 |
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Original abstract
The instabilities of the nontrivial phase elliptic solutions in a repulsive Bose-Einstein condensate (BEC) with a periodic potential are investigated. Based on the defocusing nonlinear Schr\"{o}dinger (NLS) equation with an elliptic function potential, the well-known modulational instability (MI), the more recently identified high-frequency instability, and an unprecedented -- to our knowledge -- variant of the MI, the so-called isola instability are identified numerically. Upon varying parameters of the solutions, instability transitions occur through the suitable bifurcations, such as the Hamiltonian Hope one. Specifically, (i) increasing the elliptic modulus $k$ of the solutions, we find that MI switches to the isola instability and the dominant disturbance has twice the elliptic wave's period, corresponding to a Floquet exponent $\mu=\frac{\pi}{2K(k)}$. The isola instability arises from the collision of spectral elements at the origin of the spectral plane. (ii) Upon varying $V_{0}$, the transition between MI and the high-frequency instability occurs. Differently from the MI and isola instability where the collisions of eigenvalues happen at the origin, high-frequency instability arises from pairwise collisions of nonzero, imaginary elements of the stability spectrum; (iii) In the limit of sinusoidal potential, we show that MI occurs from a collision of eigenvalues with $\mu=\frac{\pi}{2K(k)}$ at the origin; (iv) we also examine the dynamic byproducts of the instability in chaotic fields generated by its manifestation. An interesting observation is that, in addition to MI, the isola instability could also lead to dark localized events in the scalar defocusing NLS equation.
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However, here the eigenfunctions associated with such a branch don't have such local property (which is a fundamental difference in comparison to [ref] ), as shown in FIG.
The instability of near-extreme Stokes waves · From MI to isola instability · 0.93
Cites as background 1 paper
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These high-frequency instabilities have been studied analytically [ref] .
Unstable Stokes waves · Mathematical Formulation and Main Results · 0.80
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