133 research outputs found
Self trapping transition for a nonlinear impurity within a linear chain
In the present work we revisit the issue of the self-trapping dynamical
transition at a nonlinear impurity embedded in an otherwise linear lattice. For
our Schr\"odinger chain example, we present rigorous arguments that establish
necessary conditions and corresponding parametric bounds for the transition
between linear decay and nonlinear persistence of a defect mode. The proofs
combine a contraction mapping approach applied in the fully dynamical problem
in the case of a 3D-lattice, together with variational arguments for the
derivation of parametric bounds for the creation of stationary states
associated with the expected fate of the self-trapping dynamical transition.
The results are relevant for both power law nonlinearities and saturable ones.
The analytical results are corroborated by numerical computations.Comment: 16 pages, 7 figures. To be published in Journal of Mathematical
Physic
A constrained random-force model for weakly bending semiflexible polymers
The random-force (Larkin) model of a directed elastic string subject to
quenched random forces in the transverse directions has been a paradigm in the
statistical physics of disordered systems. In this brief note, we investigate a
modified version of the above model where the total transverse force along the
polymer contour and the related total torque, in each realization of disorder,
vanish. We discuss the merits of adding these constraints and show that they
leave the qualitative behavior in the strong stretching regime unchanged, but
they reduce the effects of the random force by significant numerical
prefactors. We also show that a transverse random force effectively makes the
filament softer to compression by inducing undulations. We calculate the
related linear compression coefficient in both the usual and the constrained
random force model.Comment: 4 pages, 1 figure, accepted for publication in PR
Stretching semiflexible filaments with quenched disorder
We study the effect of quenched randomness in the arc-length dependent
spontaneous curvature of a wormlike chain under tension. In the weakly bending
approximation in two dimensions, we obtain analytic results for the
force-elongation curve and the width of transverse fluctuations. We compare
quenched and annealed disorder and conclude that the former cannot always be
reduced to a simple change in the stiffness of the pure system. We also discuss
the effect of a random transverse force on the stretching response of a
wormlike chain without spontaneous curvature.Comment: 5 pages, minor changes, added references, final version as published
in PR
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Two-Component 3D Atomic Bose-Einstein Condensates Support Complex Stable Patterns
We report the computational discovery of complex, topologically charged, and spectrally stable states in three-dimensional multi-component nonlinear wave systems of nonlinear Schrödinger type. While our computations relate to two-component atomic Bose-Einstein condensates in parabolic traps, our methods can be broadly applied to high-dimensional, nonlinear systems of partial differential equations. The combination of the so-called deflation technique with a careful selection of initial guesses enables the computation of an unprecedented breadth of patterns, including ones combining vortex lines, rings, stars, and “vortex labyrinths”. Despite their complexity, they may be dynamically robust and amenable to experimental observation, as confirmed by Bogolyubov-de Gennes spectral analysis and numerical evolution simulations
Interaction of Sine-Gordon Kinks and Breathers With a Parity-Time-Symmetric Defect
The scattering of kinks and low-frequency breathers of the nonlinear sine-Gordon (SG) equation on a spatially localized parity-time-symmetric perturbation (defect) with a balanced gain and loss is investigated numerically. It is demonstrated that if a kink passes the defect, it always restores its initial momentum and energy, and the only effect of the interaction with the defect is a phase shift of the kink. A kink approaching the defect from the gain side always passes, while in the opposite case it must have sufficiently large initial momentum to pass through the defect instead of being trapped in the loss region. The kink phase shift and critical velocity are calculated by means of the collective variable method. Kink-kink (kink-antikink) collisions at the defect are also briefly considered, showing how their pairwise repulsive (respectively, attractive) interaction can modify the collisional outcome of a single kink within the pair with the defect. For the breather, the result of its interaction with the defect depends strongly on the breather parameters (velocity, frequency, and initial phase) and on the defect parameters. The breather can gain some energy from the defect and as a result potentially even split into a kink-antikink pair, or it can lose a part of its energy. Interestingly, the breather translational mode is very weakly affected by the dissipative perturbation, so that a breather penetrates more easily through the defect when it comes from the lossy side, than a kink. In all studied soliton-defect interactions, the energy loss to radiation of small-amplitude extended waves is negligible
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Dynamic instability of 3D stationary and traveling planar dark solitons
Here we revisit the topic of stationary and propagating solitonic excitations in self-repulsive three-dimensional (3D) Bose–Einstein condensates by quantitatively comparing theoretical analysis and associated numerical computations with our experimental results. Motivated by numerous experimental efforts, including our own herein, we use fully 3D numerical simulations to explore the existence, stability, and evolution dynamics of planar dark solitons. This also allows us to examine their instability-induced decay products including solitonic vortices and vortex rings. In the trapped case and with no adjustable parameters, our numerical findings are in correspondence with experimentally observed coherent structures. Without a longitudinal trap, we identify numerically exact traveling solutions and quantify how their transverse destabilization threshold changes as a function of the solitary wave speed
Formation and quench of homonuclear and heteronuclear quantum droplets in one dimension
We study the impact of beyond Lee-Huang-Yang (LHY) physics, especially due to intercomponent correlations, in the ground state and the quench dynamics of one-dimensional quantum droplets with an ab initio nonperturbative approach. It is found that the droplet Gaussian-shaped configuration arising for intercomponent attractive couplings becomes narrower for stronger intracomponent repulsion and transits towards a flat-top structure either for larger particle numbers or weaker intercomponent attraction. Additionally, a harmonic trap prevents the flat-top formation. At the balance point where mean-field interactions cancel out, we show that a correlation hole is present in the few-particle limit of LHY fluids as well as for flat-top droplets. Introducing mass imbalance, droplets experience intercomponent mixing and excitation signatures are identified for larger masses. Monitoring the droplet expansion (breathing motion) upon considering interaction quenches to stronger (weaker) attractions, we explicate that beyond LHY correlations result in a reduced velocity (breathing frequency). Strikingly, the droplets feature two-body anticorrelations (correlations) at the same position (longer distances). Our findings pave the way for probing correlation-induced phenomena of droplet dynamics in current ultracold-atom experiments
Asymptotic behavior of small solutions for the discrete nonlinear Schr\"odinger and Klein-Gordon equations
We show decay estimates for the propagator of the discrete Schr\"odinger and
Klein-Gordon equations in the form \norm{U(t)f}{l^\infty}\leq C
(1+|t|)^{-d/3}\norm{f}{l^1}. This implies a corresponding (restricted) set of
Strichartz estimates. Applications of the latter include the existence of
excitation thresholds for certain regimes of the parameters and the decay of
small initial data for relevant norms. The analytical decay estimates are
corroborated with numerical results.Comment: 13 pages, 4 figure
A Quadratic Speedup in Finding Nash Equilibria of Quantum Zero-Sum Games
Recent developments in domains such as non-local games, quantum interactive
proofs, and quantum generative adversarial networks have renewed interest in
quantum game theory and, specifically, quantum zero-sum games. Central to
classical game theory is the efficient algorithmic computation of Nash
equilibria, which represent optimal strategies for both players. In 2008, Jain
and Watrous proposed the first classical algorithm for computing equilibria in
quantum zero-sum games using the Matrix Multiplicative Weight Updates (MMWU)
method to achieve a convergence rate of iterations
to -Nash equilibria in the -dimensional spectraplex. In this
work, we propose a hierarchy of quantum optimization algorithms that generalize
MMWU via an extra-gradient mechanism. Notably, within this proposed hierarchy,
we introduce the Optimistic Matrix Multiplicative Weights Update (OMMWU)
algorithm and establish its average-iterate convergence complexity as
iterations to -Nash equilibria. This
quadratic speed-up relative to Jain and Watrous' original algorithm sets a new
benchmark for computing -Nash equilibria in quantum zero-sum games.Comment: 53 pages, 7 figures, QTML 2023 (Accepted (Long Talk)
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