共查询到20条相似文献,搜索用时 15 毫秒
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The influence of the random perturbations on the fourth-order nonlinear Schrödinger equations, , is investigated in this paper. The local well-posedness in the energy space H 2(? n ) are proved for \(p > \tfrac{{n + 4}}{{n + 2}}\), and p ≤ 2# ? 1 if n ≥ 5. Global existence is also derived for either defocusing or focusing L 2-subcritical nonlinearities.
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$iu_t + \Delta ^2 u + \varepsilon \Delta u + \lambda |u|^{p - 1} u = \dot \xi ,(t,x) \in \mathbb{R}^ + \times \mathbb{R}^n ,n \geqslant 1,\varepsilon \in \{ - 1,0, + 1\} ,$
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Paul H. Rabinowitz 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》1992,43(2):270-291
This paper concerns the existence of standing wave solutions of nonlinear Schrödinger equations. Making a standing wave ansatz reduces the problem to that of studying the semilinear elliptic equation:
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H. Mejjaoli 《Nonlinear Analysis: Theory, Methods & Applications》2010,72(3-4):1121-1139
In this paper, we introduce a class of nonlinear Schrödinger equations associated with the Dunkl operators. In this regard, we study local and global well-posedness, and the scattering theory associated with these equations. 相似文献
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We prove the existence of global classical solutions of the Cauchy-problem for nonlinear Schrödinger equations u+iAu+f(|u|2)u or u+Au+if(|u|2)u respectively. We need various growth conditions on the nonlinearity f and some restrictions on the admissible space dimension n. 相似文献
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We consider the problem $$\left\{\begin{array}{ll}-\Delta u - g(u) = \lambda u,\\ u \in H^1(\mathbb{R}^N), \int_{\mathbb{R}^N} u^2 = 1, \lambda \in \mathbb{R},\end{array}\right.$$ in dimension N ≥ 2. Here g is a superlinear, subcritical, possibly nonhomogeneous, odd nonlinearity. We deal with the case where the associated functional is not bounded below on the L 2-unit sphere, and we show the existence of infinitely many solutions. 相似文献
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We consider the nonlinear Schrödinger equations (NLS) on with random and rough initial data. By working in the framework of spaces, , we prove almost sure local well-posedness for rougher initial data than those considered in the existing literature. The main ingredient of the proof is the dispersive estimate. 相似文献
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We study the nonlinear Schrödinger equation in \(\mathbb {R}^n\) without making any periodicity assumptions on the potential or on the nonlinear term. This prevents us from using concentration compactness methods. Our assumptions are such that the potential does not change the essential spectrum of the linear operator. This results in \([0, \infty )\) being the absolutely continuous part of the spectrum. If there are an infinite number of negative eigenvalues, they will converge to 0. In each case we obtain nontrivial solutions. We also obtain least energy solutions. 相似文献
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Nakao Hayashi 《manuscripta mathematica》1986,55(2):171-190
We prove the existence of global classical solutions to the initial value problem for the nonlinear Schrödinger equation, iut–u+q(|u|2)u=0 in iut - u + (|u|2)u = in (t, x)xn for 6n11. 相似文献
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Nakao Hayashi Pavel I. Naumkin 《NoDEA : Nonlinear Differential Equations and Applications》2014,21(3):415-440
We consider the Cauchy problem for the pth order nonlinear Schrödinger equation in one space dimension $$\left\{\begin{array}{ll}iu_{t} + \frac{1}{2} u_{xx} = |u|^{p}, x \in {\bf R}, \, t > 0, \\ \qquad u(0, x) = u_{0} (x), \; x \in {\bf R},\end{array}\right.$$ where \({p > p_{s} = \frac{3 + \sqrt{17}}{2}}\) . We reveal that p = 4 is a new critical exponent with respect to the large time asymptotic behavior of solutions. We prove that if p s < p < 4, then the large time asymptotics of solutions essentially differs from that for the linear case, whereas it has a quasilinear character for the case of p > 4. 相似文献
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For elliptic equations of the form
, where the potential V satisfies
, we develop a new variational approach to construct localized bound state solutions concentrating at an isolated component of the local minimum of V where the minimum value of V can be positive or zero. These solutions give rise to standing wave solutions having a critical frequency for the corresponding nonlinear Schrödinger equations. Our method allows a fairly general class of nonlinearity f(u) including ones without any growth restrictions at large.Received: 5 July 2002, Accepted: 24 October 2002, Published online: 14 February 2003The research of the first author was supported by Grant No. 1999-2-102-003-5 from the Interdisciplinary Research Program of the KOSEF. 相似文献
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《Communications in Nonlinear Science & Numerical Simulation》2011,16(11):4232-4237
Based on two types of expanding Lie algebras of a Lie algebra G, three isospectral problems are designed. Under the framework of zero curvature equation, three nonlinear integrable couplings of the nonlinear Schröding equations are generated. With the help of variational identity, we get the Hamiltonian structure of one of them. Furthermore, we get the result that the hierarchy is also integrable in sense of Liouville. 相似文献
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Peter Y. H. Pang Hongyan Tang Youde Wang 《Calculus of Variations and Partial Differential Equations》2006,26(2):137-169
In this paper, we study blow-up solutions to the Cauchy problem of the inhomogeneous nonlinear Schrödinger equation $ \partial_t u = i ( f(x) \Delta u + \nabla f(x) \cdot \nabla u +k(x)|u|^2u) $ on ${\mathbb{R}}^2
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《Chaos, solitons, and fractals》2000,11(1-3):153-158
It is shown that in nonlinear differential–difference equations, which in the continuum limit are reduced to the nonlinear Schrödinger equation, localized excitations, reminding shock waves in liquids and gasses can propagate. Such waves may have either the conventional profile of shock waves or a shape of dark pulses evolving against a background. At the initial stages of evolution the shock waves are described by the equation ut+uux=0 and split out in a train of soliton-like pulses after the shock is developed. 相似文献
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