3,570 research outputs found
Exponentially Localized Solutions of Mel'nikov Equation
The Mel'nikov equation is a (2+1) dimensional nonlinear evolution equation
admitting boomeron type solutions. In this paper, after showing that it
satisfies the Painlev\'{e} property, we obtain exponentially localized dromion
type solutions from the bilinearized version which have not been reported so
far. We also obtain more general dromion type solutions with spatially varying
amplitude as well as induced multi-dromion solutions.Comment: 12 pages, 2 figures, to appear in Chaos, Solitons and Fractal
Localized solutions of Lugiato-Lefever equations with focused pump
Lugiato-Lefever (LL) equations in one and two dimensions (1D and 2D)
accurately describe the dynamics of optical fields in pumped lossy cavities
with the intrinsic Kerr nonlinearity. The external pump is usually assumed to
be uniform, but it can be made tightly focused too -- in particular, for
building small pixels. We obtain solutions of the LL equations, with both the
focusing and defocusing intrinsic nonlinearity, for 1D and 2D confined modes
supported by the localized pump. In the 1D setting, we first develop a simple
perturbation theory, based in the sech ansatz, in the case of weak pump and
loss. Then, a family of exact analytical solutions for spatially confined modes
is produced for the pump focused in the form of a delta-function, with a
nonlinear loss (two-photon absorption) added to the LL model. Numerical
findings demonstrate that these exact solutions are stable, both dynamically
and structurally (the latter means that stable numerical solutions close to the
exact ones are found when a specific condition, necessary for the existence of
the analytical solution, does not hold). In 2D, vast families of stable
confined modes are produced by means of a variational approximation and full
numerical simulations.Comment: 26 pages, 9 figures, accepted for publication in Scientific Report
Snakes and ladders: localized solutions of plane Couette flow
We demonstrate the existence of a large number of exact solutions of plane
Couette flow, which share the topology of known periodic solutions but are
localized in space. Solutions of different size are organized in a
snakes-and-ladders structure strikingly similar to that observed for simpler
pattern-forming PDE systems. These new solutions are a step towards extending
the dynamical systems view of transitional turbulence to spatially extended
flows.Comment: submitted to Physics Review Letter
Stabilization of localized structures by inhomogeneous injection in Kerr resonators
We consider the formation of temporal localized structures or Kerr comb
generation in a microresonator with inhomogeneities. We show that the
introduction of even a small inhomogeneity in the injected beam widens the
stability region of localized solutions. The homoclinic snaking bifurcation
associated with the formation of localized structures and clusters of them with
decaying oscillatory tails is constructed. Furthermore, the inhomogeneity
allows not only to control the position of localized solutions, but strongly
affects their stability domains. In particular, a new stability domain of a
single peak localized structure appears outside of the region of multistability
between multiple peaks of localized states. We identify a regime of larger
detuning, where localized structures do not exhibit a snaking behavior. In this
regime, the effect of inhomogeneities on localized solutions is far more
complex: they can act either attracting or repelling. We identify the pitchfork
bifurcation responsible for this transition. Finally, we use a potential well
approach to determine the force exerted by the inhomogeneity and summarize with
a full analysis of the parameter regime where localized structures and
therefore Kerr comb generation exist and analyze how this regime changes in the
presence of an inhomogeneity
Relativistic linear stability equations for the nonlinear Dirac equation in Bose-Einstein condensates
We present relativistic linear stability equations (RLSE) for
quasi-relativistic cold atoms in a honeycomb optical lattice. These equations
are derived from first principles and provide a method for computing
stabilities of arbitrary localized solutions of the nonlinear Dirac equation
(NLDE), a relativistic generalization of the nonlinear Schr\"odinger equation.
We present a variety of such localized solutions: skyrmions, solitons,
vortices, and half-quantum vortices, and study their stabilities via the RLSE.
When applied to a uniform background, our calculations reveal an experimentally
observable effect in the form of Cherenkov radiation. Remarkably, the Berry
phase from the bipartite structure of the honeycomb lattice induces a
boson-fermion transmutation in the quasi-particle operator statistics.Comment: 6 pages, 3 figure
Meromorphy and topology of localized solutions in the Thomas–MHD model
The one-dimensional MHD system first introduced by J.H. Thomas [Phys. Fluids 11, 1245 (1968)] as a model of the dynamo effect is thoroughly studied in the limit of large magnetic Prandtl number. The focus is on two types of localized solutions involving shocks (antishocks) and hollow (bump) waves. Numerical simulations suggest phenomenological rules concerning their generation, stability and basin of attraction. Their topology, amplitude and thickness are compared favourably with those of the meromorphic travelling waves, which are obtained exactly, and respectively those of asymptotic descriptions involving rational or degenerate elliptic functions. The meromorphy bars the existence of certain configurations, while others are explained by assuming imaginary residues. These explanations are tested using the numerical amplitude and phase of the Fourier transforms as probes of the analyticity properties. Theoretically, the proof of the partial integrability backs up the role ascribed to meromorphy. Practically, predictions are derived for MHD plasmas
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