Fast Soliton Scattering by Delta Impurities

Holmer, Justin; Marzuola, Jeremy L.; Zworski, Maciej

doi:10.1007/s00220-007-0261-z

Cited by 139 publications

(235 citation statements)

References 16 publications

(37 reference statements)

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“…(15) and (16) simplify for essentially real wave functions (real modulo a global complex phase) where the current I η vanishes. These equations appear to be singular at I η = 0.…”

Section: Real Solutionsmentioning

confidence: 99%

Stationary waves on nonlinear quantum graphs. II. Application of canonical perturbation theory in basic graph structures

Gnutzmann

Waltner

2016

Phys. Rev. E

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We consider exact and asymptotic solutions of the stationary cubic nonlinear Schrödinger equation on metric graphs. We focus on some basic example graphs. The asymptotic solutions are obtained using the canonical perturbation formalism developed in our earlier paper [S. Gnutzmann and D. Waltner, Phys. Rev. E 93, 032204 (2016)]. For closed example graphs (interval, ring, star graph, tadpole graph), we calculate spectral curves and show how the description of spectra reduces to known characteristic functions of linear quantum graphs in the low-intensity limit. Analogously for open examples, we show how nonlinear scattering of stationary waves arises and how it reduces to known linear scattering amplitudes at low intensities. In the short-wavelength asymptotics we discuss how genuine nonlinear effects may be described using the leading order of canonical perturbation theory: bifurcation of spectral curves (and the corresponding solutions) in closed graphs and multistability in open graphs.

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“…(15) and (16) simplify for essentially real wave functions (real modulo a global complex phase) where the current I η vanishes. These equations appear to be singular at I η = 0.…”

Section: Real Solutionsmentioning

confidence: 99%

Stationary waves on nonlinear quantum graphs. II. Application of canonical perturbation theory in basic graph structures

Gnutzmann

Waltner

2016

Phys. Rev. E

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“…(2) was studied previously by several authors. In [6,22,[25][26][27]40,41], the phenomenon of soliton scattering by the effect of the defect was observed, namely, interactions between the defect and the homogeneous medium soliton. For example, varying amplitude and velocity of the soliton, they studied how the defect is separating the soliton into two parts: one part is transmitted past the defect, the other one is captured at the defect.…”

Section: Introductionmentioning

confidence: 99%

“…For example, varying amplitude and velocity of the soliton, they studied how the defect is separating the soliton into two parts: one part is transmitted past the defect, the other one is captured at the defect. Holmer, Marzuola and Zworski [25,26] gave numerical simulations and theoretical arguments on this subject. Recently, these results were observed experimentally for a single waveguide potential [32].…”

Section: Introductionmentioning

confidence: 99%

Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential

Coz

Fukuizumi

Fibich

et al. 2008

Physica D: Nonlinear Phenomena

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We study analytically and numerically the stability of the standing waves for a nonlinear Schrödinger equation with a point defect and a power type nonlinearity. A major difficulty is to compute the number of negative eigenvalues of the linearized operator around the standing waves. This is overcome by a perturbation method and continuation arguments. Among others, in the case of a repulsive defect, we show that the standing-wave solution is stable in H 1 rad (R) and unstable in H 1 (R) under subcritical nonlinearity. Further we investigate the nature of instability: under critical or supercritical nonlinear interaction, we prove the instability by blowup in the repulsive case by showing a virial theorem and using a minimization method involving two constraints. In the subcritical radial case, unstable bound states cannot collapse, but rather narrow down until they reach the stable regime (a finite-width instability). In the nonradial repulsive case, all bound states are unstable, and the instability is manifested by a lateral drift away from the defect, sometimes in combination with a finite-width instability or a blowup instability.

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“…Within the mean field description, the dynamics of NLSE bright soliton collisions with potential barriers and wells has been widely explored (see, e.g., [65,[111][112][113][114] and Refs. therein).…”

Section: Splitting Solitary Waves At a Potential Barriermentioning

confidence: 99%

“…The behaviour of solitary waves is similar in soliton-like regimes; in particular, fast solitary wave collisions with a narrow barrier lead to smooth splitting of an incoming solitary wave into transmitted and reflected solitary waves [108,109,[114][115][116]. This behaviour is analogous to bright solitons in the NLSE scattering from a δ -function potential: it can be analytically demonstrated in such a situation that the incoming bright soliton is split into transmitted and reflected components, each of which consist mainly of a bright soliton, plus a small amount of radiation [114]. Bright solitary waves interacting with barriers much narrower than their width largely follow this prediction [108,115,116].…”

Section: Splitting Solitary Waves At a Potential Barriermentioning

confidence: 99%

Spontaneous Symmetry Breaking, Self-Trapping, and Josephson Oscillations

Malomed¹

2013

Progress in Optical Science and Photonics

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In recent years, bright soliton-like structures composed of gaseous Bose-Einstein condensates have been generated at ultracold temperature. The experimental capacity to precisely engineer the nonlinearity and potential landscape experienced by these solitary waves offers an attractive platform for fundamental study of solitonic structures. The presence of three spatial dimensions and trapping implies that these are strictly distinct objects to the true soliton solutions. Working within the zero-temperature mean-field description, we explore the solutions and stability of bright solitary waves, as well as their interactions. Emphasis is placed on elucidating their similarities and differences to the true bright soliton. The rich behaviour introduced in the bright solitary waves includes the collapse instability and symmetry-breaking collisions. We review the experimental formation and observation of bright solitary matter waves to date, and compare to theoretical predictions. Finally we discuss the current state-of-the-art of this area, including beyond-mean-field descriptions, exotic bright solitary waves, and proposals to exploit bright solitary waves in interferometry and as surface probes.

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Fast Soliton Scattering by Delta Impurities

Cited by 139 publications

References 16 publications

Stationary waves on nonlinear quantum graphs. II. Application of canonical perturbation theory in basic graph structures

Stationary waves on nonlinear quantum graphs. II. Application of canonical perturbation theory in basic graph structures

Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential

Spontaneous Symmetry Breaking, Self-Trapping, and Josephson Oscillations

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