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Non‐relativistic Eigen spectra with q‐deformed physical potentials by using the SUSY approach 下载免费PDF全文
M. Eshghi H. Mehraban M. Ghafoori 《Mathematical Methods in the Applied Sciences》2017,40(4):1003-1018
The Schrödinger equation is solved for some of the q‐deformed physical potentials within the framework of supersymmetric approach and supersymmetric Wentzel–Kramers–Brillouin approximation method. The energy levels are obtained with the corresponding normalized wave functions in terms of hypergeometric functions. The energy equations for some special cases of these potentials are discussed, and some of the thermodynamic quantities of the canonical system and the optical quantities of the two‐level system are calculated for one of the potentials. Some of the numerical results are shown, too. Copyright © 2016 John Wiley & Sons, Ltd. 相似文献
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Salmine Abdelmoula Hamadi Baklouti 《Mathematical Methods in the Applied Sciences》2017,40(15):5470-5477
We discuss bound and anti‐bound states for 2×2 matrix Schrödinger operator. We analyze the Fredholm determinant for Hamiltonians that can be represented in a multi‐channel framework. Our analysis covers the whole and the half‐line problems. We obtain some results on counting anti‐bound states between successive bound states. Copyright © 2017 John Wiley & Sons, Ltd. 相似文献
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Xing Cheng Yanfang Gao Jiqiang Zheng 《Mathematical Methods in the Applied Sciences》2016,39(8):2100-2117
We reprove global well‐posedness and scattering for the defocusing energy‐critical nonlinear Schrödinger equation in five dimensions. Inspired by the recent work of Killip and Visan, we adapt the Dodson's strategy ‘long‐time Strichartz estimate’ used in the work on mass‐critical nonlinear Schrödinger equation sets. Copyright © 2015 John Wiley & Sons, Ltd. 相似文献
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We consider a class of nonlinear Schrödinger equation in three space dimensions with an attractive potential. The nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical (in L2) nonlinearities. We study the asymptotic stability of the nonlinear bound states, i.e. periodic in time localized in space solutions. Our result shows that all solutions with small initial data, converge to a nonlinear bound state. Therefore, the nonlinear bound states are asymptotically stable. The proof hinges on dispersive estimates that we obtain for the time dependent, Hamiltonian, linearized dynamics around a careful chosen one parameter family of bound states that “shadows” the nonlinear evolution of the system. Due to the generality of the methods we develop we expect them to extend to the case of perturbations of large bound states and to other nonlinear dispersive wave type equations. 相似文献
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Jos Carlos de Albuquerque Yane Lísley Araújo Rodrigo Clemente 《Mathematical Methods in the Applied Sciences》2019,42(3):806-820
In this paper, we discuss the existence of bound and ground state solutions for a class of fractional Kirchhoff equations defined on the whole real line. The equation involves a nonlinear term with critical exponential growth in the Trudinger‐Moser sense. We deal with periodic and asymptotically periodic potential, which may change sign. We handle with the lack of compactness because of the unboundedness of the domain and the critical behavior of the nonlinearity. The main theorems are stated without the well‐known Ambrosetti‐Rabinowitz condition at infinity. 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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Brahim Alouini 《Mathematical Methods in the Applied Sciences》2021,44(1):91-103
In this article, we present, throughout two basic models of damped nonlinear Schrödinger (NLS)–type equations, a new idea to bound from above the fractal dimension of the global attractors for NLS‐type equations. This could answer the following open issue: consider, for instance, the classical one‐dimensional cubic nonlinear Schrödinger equation “How can we bound the fractal dimension of the associate global attractor without the need to assume that the external forcing term f has some decay at infinity (that is belonging to some weighted Lebesgue space)?” 相似文献
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Jun Zhang Fubiao Lin JinRong Wang 《Mathematical Methods in the Applied Sciences》2019,42(6):2069-2082
In this paper, we propose an efficient spectral‐Galerkin method based on a dimension reduction scheme for eigenvalue problems of Schrödinger equations. Firstly, we carry out a truncation from a three‐dimensional unbounded domain to a bounded spherical domain. By using spherical coordinate transformation and spherical harmonic expansion, we transform the original problem into a series of one‐dimensional eigenvalue problem that can be solved effectively. Secondly, we introduce a weighted Sobolev space to treat the singularity in the effective potential. Using the property of orthogonal polynomials in weighted Sobolev space, the error estimate for the approximate eigenvalues and corresponding eigenfunctions are proved. Error estimates show that our numerical method can achieve spectral accuracy for approximate eigenvalues and eigenfunctions. Finally, we give some numerical examples to demonstrate the efficiency of our algorithms and the correctness of the theoretical results. 相似文献
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Linghua Kong Ping Wei Yuqi Hong Peng Zhang Ping Wang 《Mathematical Methods in the Applied Sciences》2019,42(9):3222-3235
An energy‐preserving scheme is proposed for the three‐coupled nonlinear Schrödinger (T‐CNLS) equation. The T‐CNLS equation is rewritten into the classical Hamiltonian form. Then the spatial variable is discretized by using high‐order compact method to convert it into a finite‐dimensional Hamiltonian system. Next, a second‐order averaged vector field (AVF) method is employed in time which results in an energy‐preserving scheme. Some theoretical results such as convergence are investigated. In addition, it provides some numerical examples to illustrate the robustness and reliability of the theoretical results. It also explores the role of the parameters in the model and initial condition on the wave propagation. 相似文献
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《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
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In this paper, we study the multiplicity, stability, and quantitative properties of normalized standing waves for the nonlinear Schrödinger‐Poisson equation with a harmonic potential. 相似文献
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An energy‐preserving Crank–Nicolson Galerkin method for Hamiltonian partial differential equations 下载免费PDF全文
Haochen Li Yushun Wang Qin Sheng 《Numerical Methods for Partial Differential Equations》2016,32(5):1485-1504
A semidiscretization based method for solving Hamiltonian partial differential equations is proposed in this article. Our key idea consists of two approaches. First, the underlying equation is discretized in space via a selected finite element method and the Hamiltonian PDE can thus be casted to Hamiltonian ODEs based on the weak formulation of the system. Second, the resulting ordinary differential system is solved by an energy‐preserving integrator. The relay leads to a fully discretized and energy‐preserved scheme. This strategy is fully realized for solving a nonlinear Schrödinger equation through a combination of the Galerkin discretization in space and a Crank–Nicolson scheme in time. The order of convergence of our new method is if the discrete L2‐norm is employed. An error estimate is acquired and analyzed without grid ratio restrictions. Numerical examples are given to further illustrate the conservation and convergence of the energy‐preserving scheme constructed.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1485–1504, 2016 相似文献
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Local energy‐ and momentum‐preserving schemes for Klein‐Gordon‐Schrödinger equations and convergence analysis 下载免费PDF全文
Jiaxiang Cai Jialin Hong Yushun Wang 《Numerical Methods for Partial Differential Equations》2017,33(4):1329-1351
In this article, we obtain local energy and momentum conservation laws for the Klein‐Gordon‐Schrödinger equations, which are independent of the boundary condition and more essential than the global conservation laws. Based on the rule that the numerical methods should preserve the intrinsic properties as much as possible, we propose local energy‐ and momentum‐preserving schemes for the equations. The merit of the proposed schemes is that the local energy/momentum conservation law is conserved exactly in any time‐space region. With suitable boundary conditions, the schemes will be charge‐ and energy‐/momentum‐preserving. Nonlinear analysis shows LEP schemes are unconditionally stable and the numerical solutions converge to the exact solutions with order . The theoretical properties are verified by numerical experiments. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1329–1351, 2017 相似文献
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Jing Yang 《Mathematical Methods in the Applied Sciences》2015,38(17):3689-3705
We study the semilinear equation where 0 < s < 1, , V(x) is a sufficiently smooth non‐symmetric potential with , and ? > 0 is a small number. Letting U be the radial ground state of (?Δ)sU + U ? Up=0 in , we build solutions of the form for points ?j,j = 1,?,m, using a Lyapunov–Schmidt variational reduction. Copyright © 2014 John Wiley & Sons, Ltd. 相似文献
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《Mathematical Methods in the Applied Sciences》2018,41(4):1653-1660
The inverse scattering transform for the derivative nonlinear Schrödinger‐type equation is studied via the Riemann‐Hilbert approach. In the direct scattering process, the spectral analysis of the Lax pair is performed, from which a Riemann‐Hilbert problem is established for the derivative nonlinear Schrödinger‐type equation. In the inverse scattering process, N‐soliton solutions of the derivative nonlinear Schrödinger‐type equation are obtained by solving Riemann‐Hilbert problems corresponding to the reflectionless cases. Moreover, the dynamics of the exact solutions are discussed. 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
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This paper is concerned with the initial value problem for the fourth‐order nonlinear Schrödinger type equation related to the theory of vortex filament. By deriving a fundamental estimate on dyadic blocks for the fourth‐order Schrödinger through the [k,Z]‐multiplier norm method. we establish multilinear estimates for this nonlinear fourth‐order Schrödinger type equation. The local well‐posedness for initial data in with s > 1 ∕ 2 is implied by the multilinear estimates. Copyright © 2012 John Wiley & Sons, Ltd. 相似文献