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Register a new userGlobal well-posedness for the Gross-Pitaevskii equation with an angular momentum rotational term in three dimensions
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In this paper, we establish the global well-posedness of the Cauchy problem for the Gross-Pitaevskii equation with an angular momentum rotational term in which the angular velocity is equal to the isotropic trapping frequency in the space $Real^3$.
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In this paper, we establish the global well-posedness of the Cauchy problem for the Gross-Pitaevskii equation with an rotational angular momentum term in the space $Real^2$.
In this paper, we design a novel class of arbitrarily high-order structure-preserving numerical schemes for the time-dependent Gross-Pitaevskii equation with angular momentum rotation in three dimensions. Based on the idea of the scalar auxiliary variable approach which is proposed in the recent papers [J. Comput. Phys., 416 (2018) 353-407 and SIAM Rev., 61(2019) 474-506] for developing energy stable schemes for gradient flow systems, we firstly reformulate the Gross-Pitaevskii equation into an equivalent system with a modified energy conservation law. The reformulated system is then discretized by the Gauss collocation method in time and the standard Fourier pseudo-spectral method in space, respectively. We show that the proposed schemes can preserve the discrete mass and modified energy exactly. Numerical results are addressed to verify the efficiency and high-order accuracy of the proposed schemes.
The initial value problem for the $L^{2}$ critical semilinear Schrodinger equation in $R^n, n geq 3$ is considered. We show that the problem is globally well posed in $H^{s}({Bbb R^{n}})$ when $1>s>frac{sqrt{7}-1}{3}$ for $n=3$, and when $1>s> frac{-(n-2)+sqrt{(n-2)^2+8(n-2)}}{4}$ for $n geq 4$. We use the ``$I$-method combined with a local in time Morawetz estimate.
We prove global well-posedness for 3D Dirac equation with a concentrated nonlinearity.
We study the Cauchy problem for the 3D Gross-Pitaevskii equation. The global well-posedness in the natural energy space was proved by Gerard cite{Gerard}. In this paper we prove scattering for small data in the same space with some additional angular regularity, and in particular in the radial case we obtain small energy scattering.
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