Sufficient conditions of collapse for dipolar Bose‐Einstein condensate

Xie, Yingying; Mei, Liquan; Zhu, S. J.; Li, Lingfei

doi:10.1002/zamm.201700370

Cited by 4 publications

(3 citation statements)

References 23 publications

(38 reference statements)

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“…where ψ(x, t) is the complex wave envelope, |ψ(x, t)| 2 is the atomic density; g = 4π 2 N a m with the Planck constant , the total number of particles N ∈ N, the mass of single particle m > 0, the scattering length a ∈ R that can be adjusted to be positive or negative in the course of experiment; the external potential V ext represents the electromagnetic trap for the condensation [33].…”

Section: Option Pricing Model Of Quantum Originmentioning

confidence: 99%

Going beyond the Black-Scholes option pricing model: financial rogue waves in quantum mechanics

Li,

Zhu,

et al. 2024

Preprint

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This paper proposes a nonlinear alternative for the Black-Scholes option pricing model that is defined as the Gross-Pitaevskii equation. It models the controlled Brownian motion of the financial market and describes the option price wave function in terms of the compound price and time. Here, we analytically predict the existence of a family of financial rogue wave solutions of the Black-Scholes equation in the form of complex rational functions by introducing small perturbation and using symbolic computation. The obtained solutions compose of a single symmetric soliton or several solitary peaks. The existence of non-zero offset parameters provides nontrivial alterations to higher order solutions as they would decompose into first order ones that can be used to simulate the evolution of financial risks. Financial rogue wave demonstrates a condition that investors may face a huge risk or return in the financial market, which might be an actual theoretical basis for the existence of it. Our study is the first to give solid proof of the connection between option pricing and quantum mechanics. This finding opens a novel path for the excitation and control of financial waveforms of quantum mechanic footprint and extreme financial crisis.

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Section: Option Pricing Model Of Quantum Originmentioning

confidence: 99%

Going beyond the Black-Scholes option pricing model: financial rogue waves in quantum mechanics

Li,

Zhu,

et al. 2024

Preprint

View full text Add to dashboard Cite

show abstract

“…Next, we give the expression of the singular operator P, P x and Q, Q x under the new coordinate. It comes from (30) that…”

Section: An Associated Semilinear Systemmentioning

confidence: 99%

“…In addition, the researcher in [25] utilized Lagrangian coordinates to extend the previous results through lowering the common condition of the initial value for the D-G-H equation to the class of continuous differential periodic initial value, i.e., for any u 0 ∈ C 1 (R), there is some T(u 0 ) > 0 and a unique solution u ∈ C([0, T); C 1 (R)) ∩ C 1 ([0, T); C(R)). Such method is also succeeded in proving the periodic the C-H equation that lives up to the least action principle [26,27] and giving the threshold for global existence and blowup solutions [28][29][30]. Regarding the conservative solution of the C-H equation [1,31,32], Bressan and Constantin transferred the equation to the ODE system at first, in terms of a set of variables.…”

Section: Introductionmentioning

confidence: 99%

Global Dissipative Solution for an Extended Dullin–Gottwald–Holm Equation

Wang

Yan

2022

Journal of Mathematics

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The current paper devotes to the continuation of the solutions for the Dullin–Gottwald–Holm equation with forcing. Due to forcing, this equation loses the conservation of energy and cannot be regarded as the Hamiltonian system. Thus, through employing the characteristic method and exploiting the balance law, we prove the existence of a semigroup of a global dissipative solution, which is defined for each initial value u 0 ∈ H 1 ℝ and depends continuity on it.

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Global existence and long-time behavior of solutions for fully nonlocal Boussinesq equations

Zhang

Zheng

et al. 2023

era

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<abstract>In this paper, we study initial boundary value problems for the following fully nonlocal Boussinesq equation <disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ _0^{C}D_{t}^{\beta}u+(-\Delta)^{\sigma}u+(-\Delta)^{\sigma}{_0^{C}D_{t}^{\beta}}u = -(-\Delta)^{\sigma}f(u) $\end{document} </tex-math></disp-formula> with spectral fractional Laplacian operators and Caputo fractional derivatives. To our knowledge, there are few results on fully nonlocal Boussinesq equations. The main difficulty is that each term of this equation has nonlocal effect. First, we obtain explicit expressions and some rigorous estimates of the Green operators for the corresponding linear equation. Further, we get global existence and some decay estimates of weak solutions. Second, we establish new chain and Leibnitz rules concerning $ (-\Delta)^{\sigma} $. Based on these results and small initial conditions, we obtain global existence and long-time behavior of weak solutions under different dimensions $ N $ by Banach fixed point theorem.</abstract>

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Sufficient conditions of collapse for dipolar Bose‐Einstein condensate

Cited by 4 publications

References 23 publications

Going beyond the Black-Scholes option pricing model: financial rogue waves in quantum mechanics

Going beyond the Black-Scholes option pricing model: financial rogue waves in quantum mechanics

Global Dissipative Solution for an Extended Dullin–Gottwald–Holm Equation

Global existence and long-time behavior of solutions for fully nonlocal Boussinesq equations

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