共查询到20条相似文献,搜索用时 15 毫秒
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Flank D.M. Bezerra Alexandre N. Carvalho Tomasz Dlotko Marcelo J.D. Nascimento 《Journal of Mathematical Analysis and Applications》2018,457(1):336-360
We consider the Dirichlet boundary problem for semilinear fractional Schrödinger equation with subcritical nonlinear term. Local and global in time solvability and regularity properties of solutions are discussed. But our main task is to describe the connections of the fractional equation with the classical nonlinear Schrödinger equation, including convergence of the linear semigroups and continuity of the nonlinear semigroups when the fractional exponent α approaches 1. 相似文献
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The equation 1 $$ - \Delta u + qu + f(x,u) = \lambda u, u \in W^{1,2} (\mathbb{R}^N )$$ is considered, whereq is bounded below andq(x)→∞ as |x|→∞. Under appropriate conditions on the perturbation termf(x, u) it is shown that given anyr>0, (*) has an infinite sequence (λ n, r ) n ∈ N of eigenvalues, eachλ n, r being associated with an eigenfunctionu n,r which satisfies \(\smallint _{R^N } \left| {u_{n,r} } \right|^2 = r^2 \) . Information about the behaviour ofλ n, r for largen is provided. The proofs rely on the compactness of the embedding of a certain weighted Sobolov space in anL p space; this is proved in §2. 相似文献
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Rémi Carles 《PAMM》2007,7(1):1041001-1041002
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Numerical Algorithms - A nonlinear iteration scheme for nonlinear Schrödinger equation with 2-step backward differential formula (BDF) finite element method (FEM) is proposed. Energy stability... 相似文献
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The convergence of a discontinuous Galerkin method for the linear Schrödinger equation in non-cylindrical domains of ${\mathbb{R}^m}The convergence of a discontinuous Galerkin method for the linear Schr?dinger equation in non-cylindrical domains of
\mathbbRm{\mathbb{R}^m}, m ≥ 1, is analyzed in this paper. We show the existence of the resulting approximations and prove stability and error estimates
in finite element spaces of general type. When m = 1 the resulting problem is related to the standard narrow angle ‘parabolic’ approximation of the Helmholtz equation, as
it appears in underwater acoustics. In this case we investigate theoretically and numerically the order of convergence using
finite element spaces of piecewise polynomial functions. 相似文献
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《Communications in Nonlinear Science & Numerical Simulation》2011,16(6):2513-2518
The Painlevé test is performed for a new coupled nonlinear Schrödinger type equation. It is shown that this equation passes the integrability test and is P-integrable. By means of the truncated singular expansions, we construct some novel explicit solutions from the trivial zero solution. Furthermore, the traveling wave solutions are presented by direct quadrature method. 相似文献
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Arne Jensen 《Arkiv f?r Matematik》1998,36(2):363-377
Results are obtained on the scattering theory for the Schrödinger equation $i\partial _t u(t,x) = - \Delta _x u(t,x) + V(t,x)u(t,x) + F(u(t,x))$ in spacesL r (R;L q (R d )) for a certain range ofr, q, the so-called space-time scattering. In the linear case (i.e.F≡)) the relation with usual configuration space scattering is established. 相似文献
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K. Oskolkov 《Journal of Fourier Analysis and Applications》1998,4(3):341-356
Let $h(t,x): = p.v. \sum\limits_{n \in Z\backslash \left| 0 \right|} {\frac{{e^{\pi i(tn^2 + 2xn)} }}{{2\pi in}}} = \mathop {\lim }\limits_{N \to \infty } \sum\limits_{0< \left| n \right| \leqslant N} {\frac{{e^{\pi i(tn^2 + 2xn)} }}{{2\pi in}}} $ ( $(i = \sqrt { - 1;} t,x$ -real variables). It is proved that in the rectangle $D: = \left\{ {(t,x):0< t< 1,\left| x \right| \leqslant \frac{1}{2}} \right\}$ , the function h satisfies the followingfunctional inequality: $\left| {h(t,x)} \right| \leqslant \sqrt t \left| {h\left( {\frac{1}{t},\frac{x}{t}} \right)} \right| + c,$ where c is an absolute positive constant. Iterations of this relation provide another, more elementary, proof of the known global boundedness result $\left\| {h; L^\infty (E^2 )} \right\| : = ess sup \left| {h(t,x)} \right|< \infty .$ The above functional inequality is derived from a general duality relation, of theta-function type, for solutions of the Cauchy initial value problem for Schrödinger equation of a free particle. Variation and complexity of solutions of Schrödinger equation are discussed. 相似文献
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L. P. Ginzburg 《Theoretical and Mathematical Physics》1999,121(3):1641-1653
We show that under certain conditions, an electron of a many-electron system can be described by the Schrödinger equation with a local Hamiltonian. The square root of the electron density plays the role of the wave function, and the interaction with other electrons is taken into account by averaging with the exact conditional probability. The equation dictates a redefinition of the ionization energy, which is tested with the examples of the hydrogen molecule and two-electron atoms. 相似文献
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Mingfa Fei Chengming Huang Pengde Wang 《Numerical Methods for Partial Differential Equations》2020,36(2):369-393
This paper gives a rigorous error analysis of the multisymplectic Fourier pseudospectral method for the nonlinear fractional Schrödinger equation. The method preserves some intrinsic structure properties including the generalized multisymplectic conservation law. By rewriting it in a matrix form similar to that in the finite difference method, the method is shown to be convergent in the discrete L2 norm with the second-order accuracy in time and spectral accuracy in space. The key techniques in the analysis include the discrete energy method, cutoff of the nonlinearity, and a posterior bound of numerical solutions by using the inverse inequality. In a similar line, the convergence result for the symplectic Fourier pseudospectral method can also be established. Moreover, the errors in the local and global energy conservation laws of discrete systems are also investigated. Numerical tests are performed to confirm the theoretical results. 相似文献
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In this paper we focus our attention on the following nonlinear fractional Schrödinger equation with magnetic field where is a parameter, , , is the fractional magnetic Laplacian, and are continuous potentials and is a subcritical nonlinearity. By applying variational methods and Ljusternick–Schnirelmann theory, we prove existence and multiplicity of solutions for ε small. 相似文献
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K. I. Oskolkov M. A. Chakhkiev 《Proceedings of the Steklov Institute of Mathematics》2010,269(1):186-196
The function \(\psi : = \sum\nolimits_{n \in \mathbb{Z}\backslash \left\{ 0 \right\}} {{{e^{\pi i\left( {tn^2 + 2xn} \right)} } \mathord{\left/ {\vphantom {{e^{\pi i\left( {tn^2 + 2xn} \right)} } {\left( {\pi in^2 } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\pi in^2 } \right)}}} \), {t, x} ?2, is studied as a (generalized) solution of the Cauchy initial value problem for the Schrödinger equation. The real part of the restriction of ψ on the line x = 0, that is, the function \(R: = Re\psi \left| {_{x = 0} = \tfrac{2}{n}} \right.\sum\nolimits_{n \in \mathbb{N}} {\frac{{\sin \pi n^2 t}}{{n^2 }}} \), t ∈ ?, was suggested by B. Riemann as a plausible example of a continuous but nowhere differentiable function. The points are established on ?2 where the partial derivative \(\frac{{\partial \psi }}{{\partial t}}\) exists and equals ?1. These points constitute a countable set of open intervals parallel to the x-axis, with rational values of t. Thereby a natural extension of the well-known results of G.H. Hardy and J. Gerver is obtained (Gerver established that the derivative of the function R still does exist and equals ?1 at each rational point of the type \(t = \frac{a}{q}\) where both numbers a and q are odd). A basic role is played by a representation of the differences of the function ψ via Poisson’s summation formula and the oscillatory Fresnel integral. It is also proved that the number 3 4 is the sharp value of the Lipschitz-Hölder exponent of the function ψ in the variable t almost everywhere on ?2. 相似文献
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This paper investigates a Schrödinger problem with power-type nonlinearity and Lipschitz-continuous diffusion term on a bounded one-dimensional domain. Using the Galerkin method and a truncation, results from stochastic partial differential equations can be applied and uniform a priori estimates for the approximations are shown. Based on these boundedness results and the structure of the nonlinearity, it follows the unique existence of the variational solution. 相似文献