216 research outputs found
Helmholtz algebraic solitons
We report, to the best of our knowledge, the first exact analytical algebraic solitons of a generalized cubic-quintic Helmholtz equation. This class of governing equation plays a key role in photonics modelling, allowing a full description of the propagation and interaction of broad scalar beams. New conservation laws are presented, and the recovery of paraxial results is discussed in detail. The stability properties of the new solitons are investigated by combining semi-analytical methods and computer simulations. In particular, new general stability regimes are reported for algebraic bright solitons
Radiationless Travelling Waves In Saturable Nonlinear Schr\"odinger Lattices
The longstanding problem of moving discrete solitary waves in nonlinear
Schr{\"o}dinger lattices is revisited. The context is photorefractive crystal
lattices with saturable nonlinearity whose grand-canonical energy barrier
vanishes for isolated coupling strength values. {\em Genuinely localised
travelling waves} are computed as a function of the system parameters {\it for
the first time}. The relevant solutions exist only for finite velocities.Comment: 5 pages, 4 figure
Solitary waves in the Nonlinear Dirac Equation
In the present work, we consider the existence, stability, and dynamics of
solitary waves in the nonlinear Dirac equation. We start by introducing the
Soler model of self-interacting spinors, and discuss its localized waveforms in
one, two, and three spatial dimensions and the equations they satisfy. We
present the associated explicit solutions in one dimension and numerically
obtain their analogues in higher dimensions. The stability is subsequently
discussed from a theoretical perspective and then complemented with numerical
computations. Finally, the dynamics of the solutions is explored and compared
to its non-relativistic analogue, which is the nonlinear Schr{\"o}dinger
equation. A few special topics are also explored, including the discrete
variant of the nonlinear Dirac equation and its solitary wave properties, as
well as the PT-symmetric variant of the model
Modulation instability gain and localized waves by modified Frenkel-Kontorova model of higher order nonlinearity
In this paper, modulation instability and nonlinear supratransmission are
investigated in a one-dimensional chain of atoms using cubic-quartic
nonlinearity coefficients. As a result, we establish the discrete nonlinear
evolution equation by using the multi-scale scheme. To calculate the modulation
instability gain, we use the linearizing scheme. Particular attention is given
to the impact of the higher nonlinear term on the modulation instability.
Following that, full numerical integration was performed to identify modulated
wave patterns, as well as the appearance of a rogue wave. Through the nonlinear
supratransmission phenomenon, one end of the discrete model is driven into the
forbidden bandgap. As a result, for driving amplitudes above the
supratransmission threshold, the solitonic bright soliton and modulated wave
patterns are satisfied. An important behavior is observed in the transient
range of time of propagation when the bright solitonic wave turns into a
chaotic solitonic wave. These results corroborate our analytical investigations
on the modulation instability and show that the one-dimensional chain of atoms
is a fruitful medium to generate long-lived modulated waves
Breathers in inhomogeneous nonlinear lattices: an analysis via centre manifold reduction
We consider an infinite chain of particles linearly coupled to their nearest
neighbours and subject to an anharmonic local potential. The chain is assumed
weakly inhomogeneous. We look for small amplitude discrete breathers. The
problem is reformulated as a nonautonomous recurrence in a space of
time-periodic functions, where the dynamics is considered along the discrete
spatial coordinate. We show that small amplitude oscillations are determined by
finite-dimensional nonautonomous mappings, whose dimension depends on the
solutions frequency. We consider the case of two-dimensional reduced mappings,
which occurs for frequencies close to the edges of the phonon band. For an
homogeneous chain, the reduced map is autonomous and reversible, and
bifurcations of reversible homoclinics or heteroclinic solutions are found for
appropriate parameter values. These orbits correspond respectively to discrete
breathers, or dark breathers superposed on a spatially extended standing wave.
Breather existence is shown in some cases for any value of the coupling
constant, which generalizes an existence result obtained by MacKay and Aubry at
small coupling. For an inhomogeneous chain the study of the nonautonomous
reduced map is in general far more involved. For the principal part of the
reduced recurrence, using the assumption of weak inhomogeneity, we show that
homoclinics to 0 exist when the image of the unstable manifold under a linear
transformation intersects the stable manifold. This provides a geometrical
understanding of tangent bifurcations of discrete breathers. The case of a mass
impurity is studied in detail, and our geometrical analysis is successfully
compared with direct numerical simulations
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