共查询到20条相似文献，搜索用时 15 毫秒
1.
《Mathematische Nachrichten》2018,291(13):1926-1940
In this paper we prove the optimal upper bound for one‐dimensional Schrödinger operators with a nonnegative differentiable and single‐barrier potential , such that , where . In particular, if satisfies the additional condition , then for . For this result, we develop a new approach to study the monotonicity of the modified Prüfer angle function. 相似文献
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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
3.
《Numerical Methods for Partial Differential Equations》2018,34(4):1422-1454
In this article, a Fourier pseudospectral method, which preserves the conforal conservation la, is proposed for solving the damped nonlinear Schrödinger equation. Based on the energy method and the semi‐norm equivalence between the Fourier pseudospectral method and the finite difference method, a priori estimate for the new method is established, which shows that the proposed method is unconditionally convergent with order of in the discrete ‐norm, where is the time step and is the number of collocation points used in the spectral method. Some numerical results are addressed to confirm our theoretical analysis. 相似文献
4.
M. M. Alipour G. Domairry A. G. Davodi 《Numerical Methods for Partial Differential Equations》2011,27(5):1016-1025
In this work, we implement a relatively new analytical technique, the exp‐function method, for solving nonlinear equations and absolutely a special form of generalized nonlinear Schrödinger equations, which may contain high‐nonlinear terms. This method can be used as an alternative to obtain analytical and approximate solutions of different types of fractional differential equations, which is applied in engineering mathematics. Some numerical examples are presented to illustrate the efficiency and the reliability of exp‐function method. It is predicted that exp‐function method can be found widely applicable in engineering. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1016–1025, 2011 相似文献
5.
In this article we will study the initial value problem for some Schrödinger equations with Dirac-like initial data and therefore with infinite L2 mass, obtaining positive results for subcritical nonlinearities. In the critical case and in one dimension we prove that after some renormalization the corresponding solution has finite energy. This allows us to conclude a stability result in the defocusing setting. These problems are related to the existence of a singular dynamics for Schrödinger maps through the so-called Hasimoto transformation. 相似文献
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The following work is an extension of our recent paper [10]. We still deal with nonlinear eigenvalue problems of the form in a real Hilbert space ℋ︁ with a semi‐bounded self‐adjoint operator A0, while for every y from a dense subspace X of ℋ︁, B(y ) is a symmetric operator. The left‐hand side is assumed to be related to a certain auxiliary functional ψ, and the associated linear problems are supposed to have non‐empty discrete spectrum (y ∈ X). We reformulate and generalize the topological method presented by the authors in [10] to construct solutions of (∗︁) on a sphere SR ≔ {y ∈ X | ∥y∥ℋ︁ = R} whose ψ‐value is the n‐th Ljusternik‐Schnirelman level of ψ| and whose corresponding eigenvalue is the n‐th eigenvalue of the associated linear problem (∗︁∗︁), where R > 0 and n ∈ ℕ are given. In applications, the eigenfunctions thus found share any geometric property enjoyed by an n‐th eigenfunction of a linear problem of the form (∗︁∗︁). We discuss applications to elliptic partial differential equations with radial symmetry. 相似文献
7.
Mengze Gu;Hongli Sun; 《Mathematical Methods in the Applied Sciences》2024,47(1):409-418
We provide the lower bound for the ratio of the first two eigenvalues for vibrating strings with the mixed boundary condition, where the density is single-barrier defined in (−π2,π2)$$ left(-frac{pi }{2},frac{pi }{2}right) $$ and the transition point is 0. Moreover, the minimal ratio of the first two eigenvalues is attained by the constant. 相似文献
8.
Jun Zhang Shimin Lin JinRong Wang 《Mathematical Methods in the Applied Sciences》2019,42(5):1596-1608
We design and analyze an efficient numerical approach to solve the coupled Schrödinger equations with space‐fractional derivative. The numerical scheme is based on leap‐frog in time direction and Fourier method in spatial direction. The advantage of the numerical scheme is that only a linear equation needs to be solved for each time step size, and we proved that the energy and mass of space‐fractional coupled Schrödinger equations (SFCSEs) are conserved in the case of full‐discrete scheme. Moreover, we also analyze the error estimate of the numerical scheme, and numerical solutions converge with the order in L2 norm. Numerical examples are illustrated to verify the theoretical results. 相似文献
9.
Following the method of Ashbaugh-Benguria in Comm. Math. Phys. 124 (1989), 403--415; J. Differential Equations 103 (1993), 205--219, we prove an upper estimate of the arbitrary eigenvalue ratio for the regular Sturm-Liouville system. This upper estimate is sharp for Neumann boundary conditions. We also discuss the sign of and include an elementary proof of a useful trigonometric inequality first given in the aforementioned articles.
10.
Changchun Wang Jianxin Zhou 《Numerical Methods for Partial Differential Equations》2013,29(5):1778-1800
An orthogonal subspace minimization method is developed for finding multiple (eigen) solutions to the defocusing nonlinear Schrödinger equation with symmetry. As such solutions are unstable, gradient search algorithms are very sensitive to numerical errors, will easily break symmetry, and will lead to unwanted solutions. Instead of enforcing a symmetry by the Haar projection, the authors use the knowledge of previously found solutions to build a support for the minimization search. With this support, numerical errors can be partitioned into two components, sensitive versus insensitive to the negative gradient search. Only the sensitive part is removed by an orthogonal projection. Analysis and numerical examples are presented to illustrate the method. Numerical solutions with some interesting phenomena are captured and visualized by their solution profile and contour plots. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013 相似文献
11.
Tuba Gulsen Emrah Yilmaz Hikmet Koyunbakan 《Mathematical Methods in the Applied Sciences》2017,40(7):2329-2335
In this study, we solve an inverse nodal problem for p‐Laplacian Dirac system with boundary conditions depending on spectral parameter. Asymptotic formulas of eigenvalues, nodal points and nodal lengths are obtained by using modified Prüfer substitution. The key step is to apply modified Prüfer substitution to derive a detailed asymptotic estimate for eigenvalues. Furthermore, we have shown that the functions r(x) and q(x) in Dirac system can be established uniquely by using nodal parameters with the method used by Wang et al. Obtained results are more general than the classical Dirac system. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
12.
In this paper, we study the following generalized quasilinear Schrödinger equations: where N≥3, is a C1 even function, g(0) = 1, and g′(s)≥0 for all s≥0. Under some suitable conditions, we prove that the equation has a positive solution, a negative solution, and a sequence of high‐energy solutions. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
13.
We study the defocusing nonlinear Schrödinger (NLS) equation written in hydrodynamic form through the Madelung transform. From the mathematical point of view, the hydrodynamic form can be seen as the Euler–Lagrange equations for a Lagrangian submitted to a differential constraint corresponding to the mass conservation law. The dispersive nature of the NLS equation poses some major numerical challenges. The idea is to introduce a two‐parameter family of extended Lagrangians, depending on a greater number of variables, whose Euler–Lagrange equations are hyperbolic and accurately approximate NLS equation in a certain limit. The corresponding hyperbolic equations are studied and solved numerically using Godunov‐type methods. Comparison of exact and asymptotic solutions to the one‐dimensional cubic NLS equation (“gray” solitons and dispersive shocks) and the corresponding numerical solutions to the extended system was performed. A very good accuracy of such a hyperbolic approximation was observed. 相似文献
14.
Optical vortices as topological objects exist ubiquitously in nature. In this paper, we use the principle of variational method and mountain pass lemma to develop some existence theorems for the stationary vortex wave solution of a coupled nonlinear Schrödinger equations, which describe the possibility of effective waveguiding of a weak probe beam via the cross‐phase modulation‐type interaction. The main goal is to obtain a positive solution, of minimal action if possible, with all vector components not identically zero. Additionally, as demanded by beam confinement, we prove the exponential decay of the soliton amplitude at infinity. Copyright © 2017 John Wiley & Sons, Ltd. 相似文献
15.
In this paper, we study the following fractional Schrödinger equations: (1) where (?△)α is the fractional Laplacian operator with , 0≤s ≤2α , λ >0, κ and β are real parameter. is the critical Sobolev exponent. We prove a fractional Sobolev‐Hardy inequality and use it together with concentration compact theory to get a ground state solution. Moreover, concentration behaviors of nontrivial solutions are obtained when the coefficient of the potential function tends to infinity. 相似文献
16.
Yvan Martel Frank Merle 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2006,23(6):849-864
We consider the nonlinear Schrödinger equation in for any d1, with a nonlinearity such that solitary waves exist and are stable. Let Rk(t,x) be K arbitrarily given solitary waves of the equation with different speeds v1,v2,…,vK. In this paper, we prove that there exists a solution u(t) of the equation such that 相似文献
17.
François Alouges Christophe Audouze 《Numerical Methods for Partial Differential Equations》2009,25(2):380-400
In this article, we propose to solve numerically the problem of finding the smallest eigenvalues of a Hermitian operator (and the space spanned by the corresponding eigenvectors) by a gradient flow technique. This method is then applied to the Hartree‐Fock problem. Improvements are also proposed in two directions: preconditioning of the dynamical system and development of a specific flow that enables to compute directly the eigenvectors. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009 相似文献
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Claudianor O. Alves Daniel C. de Morais Filho Giovany M. Figueiredo 《Mathematical Methods in the Applied Sciences》2019,42(14):4862-4875
In this work, we prove the existence of positive solution for the following class of problems where λ>0 and is a potential satisfying some conditions. Using the variational method developed by Szulkin for functionals, which are the sum of a C1 functional with a convex lower semicontinuous functional, we prove that for each large enough λ>0, there exists a positive solution for the problem, and that, as λ→+∞, such solutions converge to a positive solution of the limit problem defined on the domain Ω=int(V?1({0})). 相似文献
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