Analysis and Computation for Ground State Solutions of Bose--Fermi Mixtures at Zero Temperature

Cai, Yongyong; Wang, Hanquan

doi:10.1137/120873820

Cited by 7 publications

(4 citation statements)

References 36 publications

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“…Similarly, the rNLSE ( 19) also conserves the mass in (15), momentum in ( 16) and the following energy as:…”

Section: The Rnlses With Global Regularizationmentioning

confidence: 99%

“…Specifically, when α = 1, i.e., f (ρ) = ρ, it is the most popular NLSE with cubic nonlinearity and also called Gross-Pitaevskii equation (GPE), especially in BEC [35,38,40]; and when α = 2, it is related to the quintic Schrödinger equation, which is regarded as the mean field limit of a Boson gas with three-body interactions and also widely used in the study of optical lattices [18,36]. When 0 < α < 1 or 1 < α < 2, it is usually stated that the NLSE with semi-smooth (or fractional) nonlinearity, which has been adapted in different applications [29,13,15,23]. For the NLSE with smooth or semi-smooth nonlinearity, i.e., α > 0, the existence and uniqueness of the Cauchy problem as well as the finite time blow-up have been widely studied [16,40].…”

Section: Introductionmentioning

confidence: 99%

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Regularized numerical methods for the nonlinear Schrödinger equation with singular nonlinearity

Bao¹,

Feng²,

Ying³

2022

Preprint

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We present different regularizations and numerical methods for the nonlinear Schrödinger equation with singular nonlinearity (sNLSE) including the regularized Lie-Trotter time-splitting (LTTS) methods and regularized Lawson-type exponential integrator (LTEI) methods. Due to the blowup of the singular nonlinearity, i.e., f (ρ) = ρ α with a fixed exponent α < 0 goes to infinity when ρ → 0 + (ρ = |ψ| 2 represents the density with ψ being the complex-valued wave function or order parameter), there are significant difficulties in designing accurate and efficient numerical schemes to solve the sNLSE. In order to suppress the round-off error and avoid blowup near ρ = 0 + , two types of regularizations for the sNLSE are proposed with a small regularization parameter 0 < 1. One is based on the local energy regularization (LER) for the sNLSE via regularizing the energy density F (ρ) = 1 α+1 ρ α+1 locally near ρ = 0 + with a polynomial approximation and then obtaining a local energy regularized nonlinear Schrödinger equation via energy variation. The other one is the global nonlinearity regularization which directly regularizes the singular nonlinearity f (ρ) = ρ α to avoid blowup near ρ = 0 + . For the regularized models, we apply the first-order Lie-Trotter time-splitting method and Lawson-type exponential integrator method for temporal discretization and combine with the Fourier pseudospectral method in space to numerically solve them. Numerical examples are provided to show the convergence of the regularized models to the sNLSE and they suggest that the local energy regularization performs better than directly regularizing the singular nonlinearity globally.

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“…Similarly, the rNLSE ( 19) also conserves the mass in (15), momentum in ( 16) and the following energy as:…”

Section: The Rnlses With Global Regularizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Regularized numerical methods for the nonlinear Schrödinger equation with singular nonlinearity

Bao¹,

Feng²,

Ying³

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…σ = 3 2 , f (ρ) = √ ρ in one dimension (1D), i.e. σ = 1 2 , and f (ρ) = ρ ln ρ in 2D; and in the mean field model for Bose-Fermi mixture [23,17], f (ρ) = ρ 2/3 , i.e. σ = 2 3 .…”

Section: Introductionmentioning

confidence: 99%

Error estimates of the time-splitting methods for the nonlinear Schrödinger equation with semi-smooth nonlinearity

Bao¹,

Wang²

2023

Preprint

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We establish error bounds of the Lie-Trotter time-splitting sine pseudospectral method for the nonlinear Schrödinger equation (NLSE) with semi-smooth nonlinearity f (ρ) = ρ σ , where ρ = |ψ| 2 is the density with ψ the wave function and σ > 0 is the exponent of the semi-smooth nonlinearity. Under the assumption of H 2 -solution of the NLSE, we prove error bounds at O(τ, respectively, and an error bound at O(τ, where h and τ are the mesh size and time step size, respectively. In addition, when 1 2 < σ < 1 and under the assumption of H 3 -solution of the NLSE, we show an error bound at O(τ σ + h 2σ ) in H 1 -norm. Two key ingredients are adopted in our proof: one is to adopt an unconditional L 2 -stability of the numerical flow in order to avoid an a priori estimate of the numerical solution for the case of 0 < σ ≤ 1 2 , and to establish an l ∞ -conditional H 1 -stability to obtain the l ∞ -bound of the numerical solution by using the mathematical induction and the error estimates for the case of σ ≥ 1 2 ; and the other one is to introduce a regularization technique to avoid the singularity of the semismooth nonlinearity in obtaining improved local truncation errors. Finally, numerical results are reported to demonstrate our error bounds.

show abstract

“…Specifically, when α = 1, i.e., f (ρ) = ρ, it is the most popular NLSE with cubic nonlinearity and also called Gross-Pitaevskii equation (GPE), especially in BEC [35,38,40]; and when α = 2, it is related to the quintic Schrödinger equation, which is regarded as the mean field limit of a Boson gas with three-body interactions and also widely used in the study of optical lattices [18,36]. When 0 < α < 1 or 1 < α < 2, it is usually stated that the NLSE with semi-smooth (or fractional) nonlinearity, which has been adapted in different applications [13,15,23,29]. For the NLSE with smooth or semi-smooth nonlinearity, i.e., α > 0, the existence and uniqueness of the Cauchy problem as well as the finite time blow-up have been widely studied [16,40].…”

Section: Introductionmentioning

confidence: 99%

Regularized Numerical Methods for the Nonlinear Schrödinger Equation with Singular Nonlinearity

Bao¹,

Yu²

2023

EAJAM

View full text Add to dashboard Cite

We present different regularizations and numerical methods for the nonlinear Schrödinger equation with singular nonlinearity (sNLSE) including the regularized Lie-Trotter time-splitting (LTTS) methods and regularized Lawson-type exponential integrator (LTEI) methods. Due to the blowup of the singular nonlinearity, i.e., f (ρ) = ρ α with a fixed exponent α < 0 goes to infinity when ρ → 0 + (ρ = |ψ| 2 represents the density with ψ being the complex-valued wave function or order parameter), there are significant difficulties in designing accurate and efficient numerical schemes to solve the sNLSE. In order to suppress the round-off error and avoid blowup near ρ = 0 + , two types of regularizations for the sNLSE are proposed with a small regularization parameter 0 < ǫ ≪ 1. One is based on the local energy regularization (LER) for the sNLSE via regularizing the energy density F (ρ) = ρ α+1 /(α + 1) locally near ρ = 0 + with a polynomial approximation and then obtaining a local energy regularized nonlinear Schrödinger equation via energy variation. The other one is the global nonlinearity regularization which directly regularizes the singular nonlinearity f (ρ) = ρ α to avoid blowup near ρ = 0 + . For the regularized models, we apply the first-order Lie-Trotter time-splitting method and Lawson-type exponential integrator method for temporal discretization and combine with the Fourier pseudospectral method in space to numerically solve them. Numerical examples are provided to show the convergence of the regularized models to the sNLSE and they suggest that the local energy regularization performs better than directly regularizing the singular nonlinearity globally.

show abstract

Analysis and Computation for Ground State Solutions of Bose--Fermi Mixtures at Zero Temperature

Cited by 7 publications

References 36 publications

Regularized numerical methods for the nonlinear Schrödinger equation with singular nonlinearity

Regularized numerical methods for the nonlinear Schrödinger equation with singular nonlinearity

Error estimates of the time-splitting methods for the nonlinear Schrödinger equation with semi-smooth nonlinearity

Regularized Numerical Methods for the Nonlinear Schrödinger Equation with Singular Nonlinearity

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