共查询到20条相似文献,搜索用时 16 毫秒
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In this paper, we study the multiplicity, stability, and quantitative properties of normalized standing waves for the nonlinear Schrödinger‐Poisson equation with a harmonic potential. 相似文献
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We consider L 2-critical focusing nonlinear Schrödinger equations with Hartree type nonlinearity $i \partial_{t} u = - \Delta u - \left(\Phi \ast |u|^2 \right) u \quad {\rm in}\, \mathbb {R}^4,$ where Φ(x) is a perturbation of the convolution kernel |x|?2. Despite the lack of pseudo-conformal invariance for this equation, we prove the existence of critical mass finite-time blowup solutions u(t, x) that exhibit the pseudo-conformal blowup rate $\| \nabla u(t) \|_{L^2_x}\sim \frac{1}{|t|} \quad {\rm as}\, t \nearrow 0.$ Furthermore, we prove the finite-codimensional stability of this conformal blow up, by extending the nonlinear wave operator construction by Bourgain and Wang (see Bourgain and Wang in Ann. Scuola Norm Sup Pisa Cl Sci (4) 25(1–2), 197–215, 1997/1998) to L 2-critical Hartree NLS. 相似文献
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In this paper, we consider the nonlinear fractional Schrödinger equations with Hartree type nonlinearity. We obtain the existence of standing waves by studying the related constrained minimization problems via applying the concentration-compactness principle. By symmetric decreasing rearrangements, we also show that the standing waves, up to translations and phases, are positive symmetric nonincreasing functions. Moreover, we prove that the set of minimizers is a stable set for the initial value problem of the equations, that is, a solution whose initial data is near the set will remain near it for all time. 相似文献
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The instability property of the standing wave uω(t, x) = eiωtφ(x) for the Klein–Gordon– Hartree equation 相似文献
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This paper is concerned with the nonlinear Klein-Gordon equations with damping term. In terms of the variational argument, the sharp conditions for blowing up and global existence are derived out by applying the potential well argument and using the concavity method. Further, the instability of the standing waves is shown. 相似文献
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In this paper, we show that the H1 solutions to the time-dependent Hartree equation
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We study stable blow-up dynamics in the generalized Hartree equation with radial symmetry, which is a Schrödinger-type equation with a nonlocal, convolution-type nonlinearity: First, we consider the -critical case in dimensions and obtain that a generic blow-up has a self-similar structure and exhibits not only the square root blowup rate , but also the log-log correction (via asymptotic analysis and functional fitting), thus, behaving similarly to the stable blow-up regime in the -critical nonlinear Schrödinger equation. In this setting, we also study blow-up profiles and show that generic blow-up solutions converge to the rescaled , a ground state solution of the elliptic equation . We also consider the -supercritical case in dimensions . We derive the profile equation for the self-similar blow-up and establish the existence and local uniqueness of its solutions. As in the NLS -supercritical regime, the profile equation exhibits branches of nonoscillating, polynomially decaying (multi-bump) solutions. A numerical scheme of putting constraints into solving the corresponding ordinary differential equation is applied during the process of finding the multi-bump solutions. Direct numerical simulation of solutions to the generalized Hartree equation by the dynamic rescaling method indicates that the is the profile for the stable blow-up. In this supercritical case, we obtain the blow-up rate without any correction. This blow-up happens at the focusing level , and thus, numerically observable (unlike the -critical case). In summary, we find that the results are similar to the behavior of stable self-similar blowup solutions in the corresponding settings for the nonlinear Schrödinger equation. Consequently, one may expect that the form of the nonlinearity in the Schrödinger-type equations is not essential in the stable formation of singularities. 相似文献
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Fbio M. Amorin Natali Ademir Pastor Ferreira 《Journal of Mathematical Analysis and Applications》2008,347(2):428-441
In the present paper we show some results concerning the orbital stability of dnoidal standing wave solutions and orbital instability of cnoidal standing wave solutions to the following Klein–Gordon equation:
utt−uxx+u−|u|2u=0.