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1.
Guixiang Xu 《Mathematical Methods in the Applied Sciences》2014,37(17):2746-2771
In this paper, we show the scattering and blow up result of the solution for some coupled nonlinear Schrödinger system with static energy less than that of the ground state in , where . We first use the Nehari manifold approach and the Schwarz symmetrization technique to construct the ground state and obtain the threshold energy of scattering solution, then use Payne–Sattinger's potential well argument and Kenig–Merle's compactness‐rigidity argument to show the aforementioned dichotomy result. As we know, it is the first attempt to show the scattering result for the coupled nonlinear Schrödinger system. Copyright © 2013 John Wiley & Sons, Ltd. 相似文献
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《Mathematical Methods in the Applied Sciences》2018,41(8):2930-2938
This paper is devoted to the Cauchy problem for the nonlinear Schrodinger equation with time‐dependent fractional damping term. We prove the local existence result, and we study the global existence and blow‐up solutions. 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 paper,we study the scattering and blow-up dichotomy result of the radial solution to nonlinear Schrodinger equation(NLS) with the combined terms iu_t+△u=-|u|~4u+|4|~(p-1)u,1+4/3p5 in energy space H~1(R~3).The threshold energy is the energy of the ground state W of the focusing,energy critical NLS,which means that the subcritical perturbation does not affect the determination of threshold,but affects the scattering and blow-up dichotomy result with subcritical threshold energy.This extends algebraic perturbation in a previous work of Miao,Xu and Zhao[Comm.Math.Phys.,318,767-808(2013)]to all mass supercritical,energy subcritical perturbation. 相似文献
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Hassan A. Zedan Seham Sh. Tantawy 《Mathematical Methods in the Applied Sciences》2009,32(9):1068-1081
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Murat Koparan 《Mathematical Methods in the Applied Sciences》2014,37(3):402-407
We perform a multiple scale analysis on the fourth order nonlinear Schrödinger equation in the Hamiltonian form together with the Hamiltonian function. We derive, as amplitude equations, Korteweg‐de Vries flow equations in the bi‐Hamiltonian form with the corresponding Hamiltonian functions. Copyright © 2013 John Wiley & Sons, Ltd. 相似文献
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This paper is concerned with the following nonlinear fractional Schrödinger equation where ε>0 is a small parameter, V(x) is a positive function, 0<s<1, and . Under some suitable conditions, we prove that for any positive integer k, one can construct a nonradial sign‐changing (nodal) solutions with exactly k maximum points and k minimum points near the local minimum point of V(x). 相似文献
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We consider the Cauchy problem for the third‐order nonlinear Schrödinger equation where and is the Fourier transform. Our purpose in this paper is to prove the large time asymptoitic behavior of solutions for the defocusing case λ > 0 with a logarithmic correction under the non zero mass condition Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
9.
Local exact controllability of the one‐dimensional NLS (subject to zero‐boundary conditions) with distributed control is shown to hold in a H1‐neighbourhood of the nonlinear ground state. The Hilbert Uniqueness Method (HUM), due to Lions, is applied to the linear control problem that arises by linearization around the ground state. The application of HUM crucially depends on the spectral properties of the linearized NLS operator which are given in detail. Copyright © 2007 John Wiley & Sons, Ltd. 相似文献
10.
Ahmet Kaçar Ömer Terzi̇oğlu 《Numerical Methods for Partial Differential Equations》2007,23(2):475-483
11.
Tengfei Zhao 《Mathematical Methods in the Applied Sciences》2016,39(12):3226-3242
In this paper, we study the nonlinear Schrödinger equation on Zoll manifolds with odd order nonlinearities. We will obtain the local well‐poesdness in the critical space . This extends the recent results in the literature to the Zoll manifolds of dimension d≥2 with general odd order nonlinearities and also partially improves the previous results in the subcritical spaces of Yang to the critical cases. Copyright © 2015 John Wiley & Sons, Ltd. 相似文献
12.
Franois Nicoleau 《Journal de Mathématiques Pures et Appliquées》2006,86(6):463-470
We study a direct and an inverse scattering problem for a pair of Hamiltonians (H(h),H0(h)) on , where H0(h)=−h2Δ and H(h)=H0(h)+V, V is a short-range potential and h is the semiclassical parameter. First, we show that if two potentials are equal in the classical allowed region for a fixed non-trapping energy, the associated scattering matrices coincide up to O(h∞) in . Then, for potentials with a regular behaviour at infinity, we study the inverse scattering problem. We show that in dimension n3, the knowledge of the scattering operators S(h), , up to O(h∞) in , and which are localized near a fixed energy λ>0, determine the potential V at infinity. 相似文献
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P. Cerejeiras N. Faustino N. Vieira 《Numerical Methods for Partial Differential Equations》2008,24(4):1181-1202
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In this paper, we consider the following perturbed nonlinear Schrödinger system with electromagnetic fields where N ≥ 3, 2 蜧 = 2N ∕ (N ? 2) is the Sobolev critical exponent; A is the real vector magnetic potential; and V (x), K(x), and H(s,t) are continuous functions. Under certain conditions on V, H, and K, we establish some new results on the existence of the least‐energy solutions (u?,v?) for small ? by using variational method. Copyright © 2012 John Wiley & Sons, Ltd. 相似文献
16.
In this paper, we consider a nonlinear sublinear Schrödinger equation at resonance in . By using bounded domain approximation technique, we prove that the problem has infinitely many solutions. Copyright © 2013 John Wiley & Sons, Ltd. 相似文献
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Ming Cheng 《Mathematical Methods in the Applied Sciences》2014,37(5):645-656
In the present paper, we consider the dissipative coupled fractional Schrödinger equations. The global well‐posedness by the contraction mapping principle is obtained. We study the long time behavior of solutions for the Cauchy problem. We prove the existence of global attractor. Copyright © 2013 John Wiley & Sons, Ltd. 相似文献
20.
Huali Zhang 《Mathematical Methods in the Applied Sciences》2016,39(14):4257-4267
In this article, the local well‐posedness of Cauchy's problem is explored for a system of quadratic nonlinear Schrödinger equations in the space Lp( R n). In a special case of mass resonant 2 × 2 system, it is well known that this problem is well posed in Hs(s≥0) and ill posed in Hs(s < 0) in two‐space dimensions. By translation on a linear semigroup, we show that the general system becomes locally well posed in Lp( R 2) for 1 < p < 2, for which p can arbitrarily be close to the scaling limit pc=1. In one‐dimensional case, we show that the problem is locally well posed in L1( R ); moreover, it has a measure valued solution if the initial data are a Dirac function. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献