410 research outputs found
Modulation instability, Fermi-Pasta-Ulam recurrence, rogue waves, nonlinear phase shift, and exact solutions of the Ablowitz-Ladik equation
We study modulation instability (MI) of the discrete constant-background wave of the Ablowitz-Ladik (A-L)
equation. We derive exact solutions of the A-L equation which are nonlinear continuations of MI at longer
times. These periodic solutions comprise a family of two-parameter solutions with an arbitrary background
field and a frequency of initial perturbation. The solutions are recurrent, since they return the field state to the
original constant background solution after the process of nonlinear evolution has passed. These solutions can
be considered as a complete resolution of the Fermi-Pasta-Ulam paradox for the A-L system. One remarkable
consequence of the recurrent evolution is the nonlinear phase shift gained by the constant background wave after
the process. A particular case of this family is the rational solution of the first-order or fundamental rogue wave.The authors acknowledge the support of the A.R.C. (Discovery
Project DP110102068). One of the authors (N.A.) is a
grateful recipient of support from the Alexander von Humboldt
Foundation (Germany)
Higher-order integrable evolution equation and its soliton solutions
We consider an extended nonlinear Schrödinger equation with higher-order odd and even terms with
independent variable coefficients. We demonstrate its integrability, provide its Lax pair, and, applying
the Darboux transformation, present its first and second order soliton solutions. The equation and its
solutions have two free parameters. Setting one of these parameters to zero admits two limiting cases:
the Hirota equation on the one hand and the Lakshmanan–Porsezian–Daniel (LPD) equation on the other
hand. When both parameters are zero, the limit is the nonlinear Schrödinger equation.A.A. and N.A. acknowledge the support of the Australian Research
Council (Discovery Project DP110102068) and also thank
the Volkswagen Foundation for financial support
Interplay between Coherence and Incoherence in Multi-Soliton Complexes
We analyze photo-refractive incoherent soliton beams and their interactions
in Kerr-like nonlinear media. The field in each of M incoherently interacting
components is calculated using an integrable set of coupled nonlinear
Schrodinger equations. In particular, we obtain a general N-soliton solution,
describing propagation of multi-soliton complexes and their collisions. The
analysis shows that the evolution of such higher-order soliton beams is
determined by coherent and incoherent contributions from fundamental solitons.
Common features and differences between these internal interactions are
revealed and illustrated by numerical examples.Comment: 4 pages, 3 figures; submitted to Physical Revie
Approach to first-order exact solutions of the Ablowitz-Ladik equation
We derive exact solutions of the Ablowitz-Ladik (A-L) equation using a special ansatz that linearly relates the real and imaginary parts of the complex function. This ansatz allows us to derive a family of first-order solutions of the A-L equation with two independent parameters. This novel technique shows that every exact solution of the A-L equation has a direct analog among first-order solutions of the nonlinear Schrödinger equation (NLSE).Two of the authors (A.A. and N.A.) acknowledge the
support of the Australian Research Council (Discovery Project
No. DP0985394). N.A. is a grateful recipient of support from
the Alexander von Humboldt Foundation (Germany)
Dynamical models for dissipative localized waves of the complex Ginzburg-Landau equation
Finite-dimensional dynamical models for solitons of the cubic-quintic complex Ginzburg-Landau equation
CGLE are derived. The models describe the evolution of the pulse parameters, such as the maximum
amplitude, pulse width, and chirp. A clear correspondence between attractors of the finite-dimensional dynamical
systems and localized waves of the continuous dissipative system is demonstrated. It is shown that stationary
solitons of the CGLE correspond to fixed points, while pulsating solitons are associated with stable limit
cycles. The models show that a transformation from a stationary soliton to a pulsating soliton is the result of
a Hopf bifurcation in the reduced dynamical system. The appearance of moving fronts kinks in the CGLE is
related to the loss of stability of the limit cycles. Bifurcation boundaries and pulse behavior in the regions
between the boundaries, for a wide range of system parameters, are found from analysis of the reduced
dynamical models. We also provide a comparison between various models and their correspondence to the
exact results
Doubly-Periodic Solutions of the Class I Infinitely Extended Nonlinear Schrodinger Equation
We present doubly-periodic solutions of the infinitely extended nonlinear
Schrodinger equation with an arbitrary number of higher-order terms and
corresponding free real parameters. Solutions have one additional free variable
parameter that allows to vary periods along the two axes. The presence of
infinitely many free parameters provides many possibilities in applying the
solutions to nonlinear wave evolution. Being general, this solution admits
several particular cases which are also given in this work.Comment: 7 pages, 5 figures. Published in Physical Review
Modulation instability, Fermi-Pasta-Ulam recurrence, rogue waves, nonlinear phase shift, and exact solutions of the Ablowitz-Ladik equation
We study modulation instability (MI) of the discrete constant-background wave of the Ablowitz-Ladik (A-L) equation. We derive exact solutions of the A-L equation which are nonlinear continuations of MI at longer times. These periodic solutions comprise
Two-breather solutions for the class I infinitely extended nonlinear Schrodinger equation and their special cases
We derive the two-breather solution of the class I infinitely extended
nonlinear Schrodinger equation (NLSE). We present a general form of this
multi-parameter solution that includes infinitely many free parameters of the
equation and free parameters of the two breather components. Particular cases
of this solution include rogue wave triplets, and special cases of
breather-to-soliton and rogue wave-to-soliton transformations. The presence of
many parameters in the solution allows one to describe wave propagation
problems with higher accuracy than with the use of the basic NLSE.Comment: 10 pages, 10 figures. Published in Nonlinear Dynamic
Sasa--Satsuma equation: Soliton on a background and its limiting cases
We present a multi-parameter family of a soliton on a background solutions to the Sasa-Satsuma equation. The solution is controlled by a set of several free parameters that control the background amplitude as well as the soliton itself. This family of solutions admits a few nontrivial limiting cases that are considered in detail. Among these special cases is the NLSE limit and the limit of rogue wave solutions
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