共查询到20条相似文献,搜索用时 0 毫秒
1.
We consider the mass-critical focusing nonlinear Schrödinger equation in the presence of an external potential, when the nonlinearity is inhomogeneous. We show that if the inhomogeneous factor in front of the nonlinearity is sufficiently flat at a critical point, then there exists a solution which blows up in finite time with the maximal (unstable) rate at this point. In the case where the critical point is a maximum, this solution has minimal mass among the blow-up solutions. As a corollary, we also obtain unstable blow-up solutions of the mass-critical Schrödinger equation on some surfaces. The proof is based on properties of the linearized operator around the ground state, and on a full use of the invariances of the equation with an homogeneous nonlinearity and no potential, via time-dependent modulations. 相似文献
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Shunlin Shen 《Journal of Functional Analysis》2021,280(8):108934
We derive rigorously the 2D periodic focusing cubic NLS as the mean-field limit of the 3D focusing quantum many-body dynamics describing a dilute Bose gas with periodic boundary condition in the x-direction and a well of infinite-depth in the z-direction. Physical experiments for these systems are scarce. We find that, to fulfill the empirical requirement for observing NLS dynamics in experiments, namely, that the kinetic energy dominates the potential energy, it is necessary to impose an extra restriction on the system parameters. This restriction gives rise to an unusual coupling constant. 相似文献
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Teng Fei Zhao 《数学学报(英文版)》2017,33(7):911-925
We consider the defocusing nonlinear Schr?dinger equations iu_t +△u =|u|~(p_u) with p being an even integer in dimensions d≥ 5. We prove that an a priori bound of critical norm implies global well-posedness and scattering for the solution. 相似文献
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We study the nonlinear Schrödinger equation with an inverse-square potential in dimensions . We consider both focusing and defocusing nonlinearities in the mass-supercritical and energy-subcritical regime. In the focusing case, we prove a scattering/blowup dichotomy below the ground state. In the defocusing case, we prove scattering in for arbitrary data. 相似文献
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Paolo Antonelli 《偏微分方程通讯》2013,38(12):2310-2328
We study the Cauchy problem for the cubic nonlinear Schrödinger equation, perturbed by (higher order) dissipative nonlinearities. We prove global in-time existence of solutions for general initial data in the energy space. In particular we treat the energy-critical case of a quintic dissipation in three space dimensions. 相似文献
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Bobby Wilson 《偏微分方程通讯》2015,40(8):1521-1542
This article explores the questions of long time orbital stability in high order Sobolev norms of plane wave solutions to the NLSE in the defocusing case. 相似文献
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Hideo Takaoka 《偏微分方程通讯》2016,41(4):732-747
In this paper, the authors discuss a priori estimates derived from the energy method to the initial value problem for the cubic nonlinear Schrödinger on the sphere S2. Exploring suitable a priori estimates, the authors prove the existence of solution for data whose regularity is s = 1/4. 相似文献
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Satoshi Masaki 《偏微分方程通讯》2013,38(12):2253-2278
We consider the Schrödinger–Poisson system in the two-dimensional whole space. A new formula of solutions to the Poisson equation is used. Although the potential term solving the Poisson equation may grow at the spatial infinity, we show the unique existence of a time-local solution for data in the Sobolev spaces by an analysis of a quantum hydrodynamical system via a modified Madelung transform. This method has been used to justify the WKB approximation of solutions to several classes of nonlinear Schrödinger equation in the semiclassical limit. 相似文献
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We study the well-posedness and long-time behavior of solution to both defocusing and focusing nonlinear Schr?dinger equations with scaling critical magnetic potentials in dimension two.In the defocusing case, and under the assumption that the initial data is radial, we prove interaction Morawetz-type inequalities and show the scattering holds in the energy space. The magnetic potential considered here is the Aharonov–Bohm potential which decays likely the Coulomb potential |x|~(-1). 相似文献
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We prove the global existence and scattering for the Hartree-type equation in H s (?3) the low regularity space s < 1. We follow the ideas in Colliander et al. (2004) to the Hartree-type nonlinearity, and also develop the theory of the classical multilinear operator modifying the L p estimate in Coifman and Meyer (1978). 相似文献
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《Communications in Nonlinear Science & Numerical Simulation》2014,19(10):3591-3599
A nonlinear Schrödinger equation that contains the time-derivative of the probability density is investigated, which is motivated by the attempt to include the recoil effect of radiation. This equation has the same stationary solutions as its linear counterpart, and these solutions are the eigen-states of the corresponding linear Hamiltonian. The equation leads to the usual continuity equation and thus maintains the normalization of the wave function. For the non-stationary solutions, numerical calculations are carried out for the one-dimensional infinite square-well potential (1D ISWP) and for several time-dependent potentials that tend to the former as time increases. Results show that for various initial states, the wave function always evolves into some eigen-state of the corresponding linear Hamiltonian of the 1D ISWP. For a small time-dependent perturbation potential, solutions present the process similar to the spontaneous transition between stationary states. For a periodical potential with an appropriate frequency, solutions present the process similar to the stimulated transition. This nonlinear Schrödinger equation thus presents the state evolution similar to the wave-function reduction. 相似文献
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I show that H 1 solutions of the nonlinear Schrödinger equation which are incoming converge to a soliton, in the radial case. 相似文献
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We study here the binding of atoms and molecules and the stability of general molecular systems including molecular ions. This is the first paper of a series devoted to the study of these general problems. We obtain here a general necessary and sufficient condition for the stability of general molecular ststem in the context of thomasz-Fermi-Von Weiasäcker, Thomas-Fermi-Dirac-Von Weizsaäcker, Hartree or Hartree-Fock theories SUMARY OF PART 1 1.Introduction. II.Presentation of the models III.Diatomic molecular systems and hartree-Fock theory IV.Diatomic molecular systems and Hartree or Thomas-Fermi theories V.General molecular systems Appendix 1: Hartree-Fock models when Z > N ― 1 Appendix 2: Dichotomy yields equal Lagrange multipliers Appendix 3: The problem at infinty for the TRDW model 相似文献
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Ze LI 《Frontiers of Mathematics in China》2020,15(5):923
We prove the asymptotic stability of solitary waves to 1D nonlinear Schrödinger equations in the subcritical case with symmetry and spectrum assumptions. One of the main ideas is to use the vector fields method developed by S. Cuccagna, V. Georgiev, and N. Visciglia [Comm. Pure Appl. Math., 2013, 6: 957–980] to overcome the weak decay with respect to t of the linearized equation caused by the one dimension setting and the weak nonlinearity caused by the subcritical growth of the nonlinearity term. Meanwhile, we apply the polynomial growth of the high Sobolev norms of solutions to 1D Schrödinger equations obtained by G. Staffilani [Duke Math. J., 1997, 86(1): 109–142] to control the high moments of the solutions emerging from the vector fields method. 相似文献
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Yongshuai Zhang Xiangxing Tao Tengteng Yao Jingsong He 《Studies in Applied Mathematics》2020,145(4):812-827
Based on the inverse scattering method, the formulae of one higher-order pole solitons and multiple higher-order poles solitons of the nonlinear Schrödinger equation (NLS) equation are obtained. Their denominators are expressed as , where is a matrix frequently constructed for solving the Riemann-Hilbert problem, and the asterisk denotes complex conjugate. We take two methods for proving is invertible. The first one shows matrix is equivalent to a self-adjoint Hankel matrix , proving . The second one considers the block-matrix form of , proving . In addition, we prove that the dimension of is equivalent to the sum of the orders of pole points of the transmission coefficient and its diagonal entries compose a set of basis. 相似文献
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Under investigation in this paper is the integrability and dark soliton solutions for a fifth-order variable coefficients nonlinear Schrödinger equation, which is used in an inhomogeneous optical fiber. Bilinear forms, Lax pair and infinitely-many conservation laws are obtained under an integrable constraint. Dark one-, two- and N-soliton solutions are constructed via the Hirota bilinear method. 相似文献
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This paper is the first in a series papers devoted to the study of the rigorous derivation of the nonlinear Schrödinger (NLS) equation as well as other related systems starting from a model coming from the gravity‐capillary water wave system in the long‐wave limit. Our main goal is to understand resonances and their effects on having the nonlinear Schrödinger approximation or modification of it or having other models to describe the limit equation. In this first paper, our goal is not to derive NLS but to allow the presence of an arbitrary sequence of frequencies around which we have a modulation and prove local existence on a uniform time. This yields a new class of large data for which we have a large time of existence. © 2012 Wiley Periodicals, Inc. 相似文献
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In this article, we construct a splitting method for nonlinear stochastic equations of Schrödinger type. We approximate the solution of our problem by the sequence of solutions of two types of equations: one without stochastic integral term, but containing the Laplace operator and the other one containing only the stochastic integral term. The two types of equations are connected to each other by their initial values. We prove that the solutions of these equations both converge strongly to the solution of the Schrödinger type equation. 相似文献