共查询到20条相似文献,搜索用时 15 毫秒
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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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A second-order Schrödinger differential operator of parabolic type is considered, for which an explicit form of a fundamental solution is derived. Such operators arise in many problems of physics, and the fundamental solution plays the role of the Feynman propagation function. 相似文献
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Thomas Bartsch 《Journal of Fixed Point Theory and Applications》2013,13(1):37-50
We consider the system ${-\Delta{u}_{j} + a(x)u_{j} = \mu_{j}u^{3}_{j} + \beta \sum_{k \neq j} u^{2}_{k}u_{j}}$ , u j > 0, j = 1, . . . , n, on a possibly unbounded domain ${\Omega \subset \mathbb{R}^{N}, N \leq 3}$ , with Dirichlet boundary conditions. The system appears in nonlinear optics and in the analysis of mixtures of Bose–Einstein condensates. We consider the self-focussing (attractive self-interaction) case ${\mu_{1}, \ldots, \mu_{n} > 0}$ and take ${\beta \in \mathbb{R}}$ as bifurcation parameter. There exists a branch of positive solutions with uj/uk being constant for all ${j, k \in \{1, \ldots, n\}}$ . The main results are concerned with the bifurcation of solutions from this branch. Using a hidden symmetry we are able to prove global bifurcation even when the linearization has even-dimensional kernel (which is always the case when n > 1 is odd). 相似文献
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In this article, a compact finite difference scheme for the coupled nonlinear Schrödinger equations is studied. The scheme is proved to conserve the original conservative properties. Unconditional stability and convergence in maximum norm with order O(τ2 + h4) are also proved by the discrete energy method. Finally, numerical results are provided to verify the theoretical analysis. 相似文献
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Annali di Matematica Pura ed Applicata (1923 -) - We study the following system of nonlinear Schrödinger equations: $$\begin{aligned} \left\{ \begin{array}{l} -\varepsilon ^2\Delta u +a(x) u =... 相似文献
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Giovany M. Figueiredo Marcelo F. Furtado 《NoDEA : Nonlinear Differential Equations and Applications》2008,15(3):309-334
We consider the quasilinear system
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Lu(t)+(u,F)g(t)=f(t), tS. , ( F, g). .
The authors wish to thank Professor Yu. A. Rozanov for his help and discussions. 相似文献 9.
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The influence of the random perturbations on the fourth-order nonlinear Schrödinger equations, 相似文献
$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\} ,$ 11.
E. A. Kopylova 《Proceedings of the Steklov Institute of Mathematics》2010,270(1):165-171
We strengthen the known Agmon-Jensen-Kato decay of the resolvent for a special case of the Schrödinger equation in arbitrary dimension n ≥ 1. The decay is of crucial importance in applications to linear and nonlinear hyperbolic PDEs. 相似文献
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This paper discusses the weakly coupled nonlinear Schrödinger equations in the supercritical case. With the best constant of the Gagliardo-Nirenberg inequality, we derive a sufficient condition for the global existence of solutions; this condition is expressed in terms of stationary solutions (nonlinear ground state). 相似文献
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By virtue of numerical arguments we study a bifurcation phenomenon occurring for a class of minimization problems associated with the so-called quasi-linear Schrödinger equation, object of various investigations in the last two decades. 相似文献
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In this paper we will study the stability properties of self-similar solutions of $1$ D cubic NLS equations with time-dependent coefficients of the form
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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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