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In this article, a generalized Darboux transformation for the fourth-order nonlinear Schrödinger equation is constructed in terms of Darboux matrix method. Subsequently, breathers and the Nth-order rogue wave solutions of this equation are explicitly given in the light of the obtained Darboux transformation. Finally, we concretely discuss the dynamics of the obtained rogue waves, which are also demonstrated by some figures. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the famous Hasimoto's transformation by showing that the dynamics of a closed unidimensional vortex filament embedded in a three-dimensional manifold M of constant curvature, gives rise under Hasimoto's transformation to the nonlinear Schrödinger equation. We also give a natural interpretation of the function ψ introduced by Hasimoto in terms of moving frames associated to a natural complex bundle over the filament. [ABSTRACT FROM AUTHOR]

We provide some explicit examples wherein the Schrödinger equation for the Morse potential remains exactly solvable in a position-dependent mass background. Furthermore, we show how in such a context, the map from the full line (-∞, ∞) to the half-line (0, ∞) may convert an exactly solvable Morse potential into an exactly solvable Coulomb one. This generalizes a well-known property of constant-mass problems. [ABSTRACT FROM AUTHOR]

The situation of a particle interacting with two electric fields, one static and the other periodic in time, may be of interest in the study of the conductance of nanostructures. The static field corresponds to a bias voltage applied between the two electrodes, while the oscillatory field is due to laser illumination. The problem can be described by a variety of Hamiltonians, some time periodic, others not. Following some recent investigations about the solution of the one-dimensional wave equation for the time-dependent linear potential, it is shown that there are solutions of the time-dependent Schrödinger equation with a periodic Hamiltonian that do not follow the usual Floquet pattern of being the product of an energy-dependent exponential by a time-periodic factor. We show how to relate the two types of solutions, those conforming with the Floquet ansatz and the non-Floquet ones. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006 [ABSTRACT FROM AUTHOR]

We set up a reality condition for Darboux transformations of time-dependent Schrödinger equations (TDSE) with position-dependent (effective) mass. The reality condition guarantees the potential in the transformed TDSE to be real-valued and hence physical. This paper is a sequel of our former one.36 [ABSTRACT FROM AUTHOR]

In this short topical review, we revisit a number of works on the pattern-forming dynamical instabilities of Bose­Einstein condensates in one- and two-dimensional settings. In particular, we illustrate the trapping conditions that allow the reduction of the three-dimensional, mean field description of the condensates (through the Gross­Pitaevskii equation) to such lower dimensional settings, as well as to lattice settings. We then go on to study the modulational instability in one dimension and the snaking/transverse instability in two dimensions as typical examples of long-wavelength perturbations that can destabilize the condensates and lead to the formation of patterns of coherent struc- tures in them. Trains of solitons in one dimension and vortex arrays in two dimensions are prototypical examples of the resulting nonlinear waveforms, upon which we briefly touch at the end of this review. [ABSTRACT FROM AUTHOR]

We consider the initial-value problem for the radially symmetric nonlinear Schrödinger equation with cubic nonlinearity (NLS) in d = 2 and 3 space dimensions. To approximate smooth solutions of this problem, we construct and analyze a numerical method based on a standard Galerkin finite element spatial discretization with piecewise linear, continuous functions and on an implicit Crank­Nicolson type time-stepping procedure. We then equip this scheme with an adaptive spatial and temporal mesh refinement mechanism that enables the numerical technique to approximate well singular solutions of the NLS equation that blow up at the origin as the temporal variable t tends from below to a finite value t[this equation cannot be represented in Ascii text]. For the blow-up of the amplitude of the solution we recover numerically the well-known rate (t[this equation cannot be represented in Ascii text] - t)-1/2 for d = 3. For d = 2 our numerical evidence supports the validity of the log log law [1n 1n [this equation cannot be represented in Ascii text]/(t[this equation cannot be represented in Ascii text] - t)] 1/2 for t extremely close to t[this equation cannot be represented in Ascii text]. The scheme also approximates well the details of the blow-up of the phase of the solution at the origin as t → t[this equation cannot be represented in Ascii text]. [ABSTRACT FROM AUTHOR]

A Hamiltonian describing four bosons that move on a lattice and interact by means of pair zero-range attractive potentials is considered. A stronger version of the Hunziker–Van Vinter–Zhislin theorem on the essential spectrum is established. It is proved that the set of eigenvalues lying to the left of the essential spectrum is finite for any interaction energy of two bosons and is empty if this energy is sufficiently small. [ABSTRACT FROM AUTHOR]

In this paper we consider the study of positive solutions of¶¶ $ -\varepsilon^2\Delta u+\lambda u=f(x,u)\quad {\rm on}\quad \mathbb{R}^N, $ ¶¶where ε is a small parameter, λ>0 and f is an appropriate function. Here we find multi-peak solutions exhibiting concentration at any prescribed "stable" set of zeroes of the field¶¶ $ {\cal S}(P)=\int\limits_{\mathbb{R}^N}\left[\nabla_xf(P,U_P(y))\cdot y\right]\nabla U_P(y)dy,\quad P\in \mathbb{R}^N, $ ¶¶where U P is the unique radial solution of the limit equation¶¶ $ -\Delta U_P+\lambda U_P=f(P,U_P)\quad {\rm on} \quad \mathbb{R}^N. $ ¶¶Conversely, we show that the points at which a sequence of multi-peak solutions concentrate must be zeroes of the field $ {\cal S} $ . [ABSTRACT FROM AUTHOR]

We unify two distinct methods of the global analysis for the nonlinear Schrödinger equations, namely those in the Sobolev spaces and in the weighted spaces. Thus we can deal with various sums of power nonlinearies $ |u|^{p-1} u $ for 1+2/ n< p< $ \infty $ , since the former works for $ p\ge 1+4/n $ , while the latter for 1+2/ n< p $ \le 1+4/n $ . Even for a single power, our result is much simpler and slightly better than the previous ones as to restriction on the initial data. Moreover, we extend the result to the maximal regularity, thereby obtaining scattering at the lower critical value $ p=1+8/(\sqrt{n^2+4n+36}+n+2)\quad \textrm{for}\quad n\ge 4 $ . We also show the asymptotic completeness in $ {\cal F}H^1 $ without smallness for $ p\ge 1+8/(\sqrt{n^2+12n+4}+n-2) $ and any $ n\in\mathbb{N} $ . [ABSTRACT FROM AUTHOR]

An exponentially‐fitted Runge–Kutta method for the numerical integration of the radial Schrödinger equation is developed. Theoretical and numerical results obtained for the well known Woods–Saxon potential show the efficiency of the new method. [ABSTRACT FROM PUBLISHER]

We show he orbital stability of the standing waves in nonlinear Schrödinger equations with a class of unbounded potentials by the method of Cazenave and Lions [3] as well as a compactness lemma. [ABSTRACT FROM AUTHOR]

We are interested in positive entire solutions of the nonlinear Schrödinger equation $-\Delta u+(\lambda a(x)+1)u = u^p$ where a? 0 has a potential well and p > 1 is subcritical. Using variational methods we prove the existence of multiple positive solutions which localize near the potential well int( a -1(0)) for $\lambda$ large. [ABSTRACT FROM AUTHOR]

The first boundary value problem for a nonlinear Schrödinger equation is investigated. The conditional convergence and stability on the initial data of the explicit three-level difference scheme of DuFort­Frankel type in C and W21 norms are proved. Grid analogues of energy conservation laws and grid multiplicative inequalities are used. [ABSTRACT FROM AUTHOR]

Two discrete-time orthogonal spline collocation schemes are formulated and analyzed for solving the linear time-dependent Schrödinger equation in two space variables. These are Crank­Nicolson and alternating direction implicit (ADI) schemes employing C1 piecewise polynomial spaces of arbitrary degree ≥ 3 in each space variable. The stability of the schemes is examined and optimal order a priori L2- and H1-error estimates at each time step are derived. Parallel implementation of the ADI scheme is discussed. [ABSTRACT FROM AUTHOR]

In this paper a two-level preconditioner is considered for the Schrödinger-type singular operator on a uniform grid on a rectangle. The construction of such preconditioners for a positive definite operator is well defined provided that a restriction of the operator into any subdomain of computational domain is at least positive semidefinite. It is shown that although local matrices approximating the Schrödinger operator on some subdomains are indefinite it is still possible to develop a multilevel preconditioner, which is spectrally equivalent to the operator of the problem. First, explicit estimates of condition number are established for a two-level preconditioner. The main idea of the two-level estimate is to analyze a superelement-based method for a problem with indefinite local superelement matrices, modify the main diagonals of the local matrices so that they are positive definite and still sum up to the same global matrix. This modification is for theoretical purposes only and does not affect the computational method itself. A generalization of the usual two-level method is considered in which different bilinear forms are defined on the coarse and define levels. Then, the two-level method can be extended to a multilevel one using the standard theory of multilevel methods with inner iterative procedures. [ABSTRACT FROM AUTHOR]

An iterative method is proposed to solve the Schrödinger eigenvalue problem in a wavelet framework. Orthonormal wavelets are used to represent the corresponding operator as a sparse band matrix. This representation, called the nonstandard (NS) form, is obtained by means of the Beylkin­Coifman­Rokhlin (BCR) algorithm and simplifies the numerical calculations. Problems due to the one-dimensional mathematical model and to the discretization process receive special attention. [ABSTRACT FROM AUTHOR]

In this paper we establish the existence of a solution of the Schrödinger equation (1). The existence of a solution is obtained by a variational method based on the critical point theorem 2.4 from [3]. [ABSTRACT FROM AUTHOR]

We consider nonlinear magnetic Schrödinger equations under assumptions that imply the Palais–Smale condition for the corresponding functional and prove some results on the existence and multiplicity of solutions vanishing at infinity. [ABSTRACT FROM AUTHOR]