Bound and resonance states of the nonlinear Schrödinger equation in simple model systems

Witthaut, Dirk; Mossmann, S.; Korsch, Hans Jürgen

doi:10.1088/0305-4470/38/8/013

Cited by 51 publications

(64 citation statements)

References 9 publications

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“…In the 3D case the boundary condition used to obtain the reflection function has to be suitably changed, since s-wave 3D scattering can be considered as equivalent to reflection without transmission. Following this method one can also obtain the eigen value equations (7)(8)(9)(10)(11)(12). In a way, the procedure based on the reflection function is more comprehensive since it automatically incorporates both bound state and scattering situation in a unified way.…”

Section: Algebraic Equations Governing the Eigen Valuesmentioning

confidence: 99%

Conditions governing the generalisation of threshold bound states by N attractive delta potentials in one and three dimensions

2013

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Abstract:This paper proves that for N attractive delta function potentials the number of bound states (N ) satisfies, and is 0 ≤ N ≤ N in three dimensions (3D). Algebraic equations are obtained to evaluate the bound states generated by N attractive delta potentials. In particular, in the case of N attractive delta function potentials having same separation between adjacent wells and having the same strength λV, the parameter g=λVa governs the number of bound states. For a given N in the range 1-7, both in 1D and 3D cases the numerical values of g , where n=1,2,..N are obtained. When g=g , N ≤ n where N includes one threshold energy bound state. Furthermore, g are the roots of the Nth order polynomial equations with integer coefficients. Based on our numerical calculations up to N=40, even when N becomes large, 0 ≤ ≤ 4 and Σ N 2 and this result is expected to be generally valid. Thus, for g > 4 there will be no threshold or zero energy bound state, and if g≈ 2 for a given large N, the number of bound states will be approximately N/2. The empirical formula g = 4/[1 + ((N 0 − )/β)] gives a good description of the variation of g as a function of n . This formula is useful in estimating the number of bound states for any N and g both in 1D and 3D cases.

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Section: Algebraic Equations Governing the Eigen Valuesmentioning

confidence: 99%

Conditions governing the generalisation of threshold bound states by N attractive delta potentials in one and three dimensions

2013

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“…First, we recall that substituting (16) in (1) is equivalent to demanding that ψ is continuous at x = a and that ψ ′ (a + ) = ψ ′ (a − ) + zψ(a), where ψ ′ (a +/− ) stands for the right/left derivative of ψ at x = a. Next, we use (17) to obtain a perturbative expression for the solution ξ k of (2) in the interval [0, a) and use the continuity of ψ at x = 0 and the above matching condition for ψ ′ to extend it to [a, 1].…”

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confidence: 99%

“…Eqs. (19) and (20) provide a reliable description of the NSSs of (16) provided that the right-hand side of (19) is much smaller than k 2 . This implies that 0…”

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confidence: 99%

Nonlinear Spectral Singularities for Confined Nonlinearities

Mostafazadeh

2013

Phys. Rev. Lett.

142

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We introduce a notion of spectral singularity that applies for a general class of nonlinear Schrödinger operators involving a confined nonlinearity. The presence of the nonlinearity does not break the parity-reflection symmetry of spectral singularities but makes them amplitude-dependent. Nonlinear spectral singularities are, therefore, associated with a resonance effect that produces amplified waves with a specific amplitude-wavelength profile. We explore the consequences of this phenomenon for a complex δ-function potential that is subject to a general confined nonlinearity. Pacs numbers: 03.65. Nk, 42.25.Bs, 24.30.Gd Introduction.-A spectral singularity of a complex scattering potential is a mathematical concept introduced and studied by mathematicians for more than half a century [1,2]. This concept that applies only for non-real scattering potentials, was recently shown to have an intriguing physical interpretation [3]: It corresponds to the energy of a scattering state whose reflection and transmission coefficients diverge. This observation has motivated identifying spectral singularities with certain zerowidth resonances and led to the study of their physical implications [3][4][5]. In optics, a spectral singularity gives rise to lasing at the threshold gain [6]. Its time-reversal corresponds to a coherent perfect absorption (CPA) of light [7] that is also called anti-lasing. The observation of the latter reported in [8] may be considered as an experimental evidence for the physical relevance of spectral singularities.Once one creates a spectral singularity in an optically active material [3,4], it begins amplifying the background noise and functions as a laser. One may argue that a realistic treatment of this phenomenon should also take into account the nonlinearities associated with the production of high-intensity radiation in the gain region. This provides our main physical motivation for the study of the meaning and behavior of spectral singularities for nonlinear operators, a task that to the best of our knowledge has not been considered previously [9]. The search for an appropriate nonlinear extension of the notion of a spectral singularity is particularly important not only because it has been an open mathematical problem for several decades, but also because it can pave the way for the discovery of the nonlinear analogs of the interesting physical phenomena such as threshold lasing [6], antilasing [8], and CPA-lasers [7] in areas other than optics.Motivated by the physical meaning of a spectral singularity of a linear Schrödinger operator [3], in this letter we propose a definition for a spectral singularity of the nonlinear Schrödinger operators H γ of the form

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“…The first derivative of the wave function ψ is discontinuous at x = 2πn (cf. the studies [2,20,54,56,67] of a NLSE for a single delta-potential) 6) whereas the wave function itself is continuous. The discontinuity of the delta-comb potential does not affect the area-preserving quality of the flow (ψ(…”

Section: Solutions Of the Nonlinear Schrödinger Equationmentioning

confidence: 99%

The Nonlinear Schrödinger Equation for the Delta-Comb Potential: Quasi-Classical Chaos and Bifurcations of Periodic Stationary Solutions

Witthaut¹,

Rapedius²,

Korsch³

2021

JNMP

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The nonlinear Schrödinger equation is studied for a periodic sequence of delta-potentials (a delta-comb) or narrow Gaussian potentials. For the delta-comb the time-independent nonlinear Schrödinger equation can be solved analytically in terms of Jacobi elliptic functions and thus provides useful insight into the features of nonlinear stationary states of periodic potentials. Phenomena well-known from classical chaos are found, such as a bifurcation of periodic stationary states and a transition to spatial chaos. The relation to new features of nonlinear Bloch bands, such as looped and period doubled bands, are analyzed in detail. An analytic expression for the critical nonlinearity for the emergence of looped bands is derived. The results for the delta-comb are generalized to a more realistic potential consisting of a periodic sequence of narrow Gaussian peaks and the dynamical stability of periodic solutions in a Gaussian comb is discussed.

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Bound and resonance states of the nonlinear Schrödinger equation in simple model systems

Cited by 51 publications

References 9 publications

Conditions governing the generalisation of threshold bound states by N attractive delta potentials in one and three dimensions

Conditions governing the generalisation of threshold bound states by N attractive delta potentials in one and three dimensions

Nonlinear Spectral Singularities for Confined Nonlinearities

The Nonlinear Schrödinger Equation for the Delta-Comb Potential: Quasi-Classical Chaos and Bifurcations of Periodic Stationary Solutions

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