805 research outputs found
Rogue wave solutions for an inhomogeneous fifth-order nonlinear Schrodinger equation from Heisenberg ferromagnetism
In this paper, generalized Darboux transformation for an inhomogeneous fifth-order nonlinear Schrodinger equation from Heisenberg ferromagnetism are constructed according to which rouge wave solutions of the equation are obtained. Influences of equation parameter on the evolution of rogue waves are discussed. With the aid of Mathematica, some special solutions are graphically illustrated which could help to better understand the evolution of rogue waves
Exact Solutions for Domain Walls in Coupled Complex Ginzburg - Landau Equations
The complex Ginzburg Landau equation (CGLE) is a ubiquitous model for the
evolution of slowly varying wave packets in nonlinear dissipative media. A
front (shock) is a transient layer between a plane-wave state and a zero
background. We report exact solutions for domain walls, i.e., pairs of fronts
with opposite polarities, in a system of two coupled CGLEs, which describe
transient layers between semi-infinite domains occupied by each component in
the absence of the other one. For this purpose, a modified Hirota bilinear
operator, first proposed by Bekki and Nozaki, is employed. A novel
factorization procedure is applied to reduce the intermediate calculations
considerably. The ensuing system of equations for the amplitudes and
frequencies is solved by means of computer-assisted algebra. Exact solutions
for mutually-locked front pairs of opposite polarities, with one or several
free parameters, are thus generated. The signs of the cubic gain/loss, linear
amplification/attenuation, and velocity of the coupled-front complex can be
adjusted in a variety of configurations. Numerical simulations are performed to
study the stability properties of such fronts.Comment: Journal of the Physical Society of Japan, in pres
Sharp well-posedness results for the generalized Benjamin-Ono equation with high nonlinearity
We establish the local well-posedness of the generalized Benjamin-Ono
equation in
, for and without smallness assumption on the
initial data. The condition is known to be sharp since the solution
map is not of class on for
. On the other hand, in the particular case of the cubic
Benjamin-Ono equation, we prove the ill-posedness in ,
Nondegeneracy and Stability of Antiperiodic Bound States for Fractional Nonlinear Schr\"odinger Equations
We consider the existence and stability of real-valued, spatially
antiperiodic standing wave solutions to a family of nonlinear Schr\"odinger
equations with fractional dispersion and power-law nonlinearity. As a key
technical result, we demonstrate that the associated linearized operator is
nondegenerate when restricted to antiperiodic perturbations, i.e. that its
kernel is generated by the translational and gauge symmetries of the governing
evolution equation. In the process, we provide a characterization of the
antiperiodic ground state eigenfunctions for linear fractional Schr\"odinger
operators on with real-valued, periodic potentials as well as a
Sturm-Liouville type oscillation theory for the higher antiperiodic
eigenfunctions.Comment: 46 pages, 2 figure
Dynamics of Defects in the Vector Complex Ginzburg-Landau Equation
Coupled Ginzburg-Landau equations appear in a variety of contexts involving
instabilities in oscillatory media. When the relevant unstable mode is of
vectorial character (a common situation in nonlinear optics), the pair of
coupled equations has special symmetries and can be written as a vector complex
Ginzburg-Landau equation. Dynamical properties of localized structures of
topological character in this vector-field case are considered. Creation and
annihilation processes of different kinds of vector defects are described, and
some of them interpreted in theoretical terms. A transition between different
regimes of spatiotemporal dynamics is described.Comment: 35 pages of LATeX, using the elsart macros. Includes 17 (large)
figures. Related material, including movies and higher resolution figures,
available at
http://www.imedea.uib.es/PhysDept/Nonlinear/research_topics/Vcgl2
Few-cycle optical rogue waves: Complex modified Korteweg-de Vries equation
In this paper, we consider the complex modified Korteweg–de Vries (mKdV) equation as a model of few-cycle optical pulses. Using the Lax pair, we construct a generalized Darboux transformation and systematically generate the first-, second-, and third-order rogue wave solutions and analyze the nature of evolution of higher-order rogue waves in detail. Based on detailed numerical and analytical investigations, we classify the higher-order rogue waves with respect to their intrinsic structure, namely, fundamental pattern, triangular pattern, and ring pattern. We also present several new patterns of the rogue wave according to the standard and nonstandard decomposition. The results of this paper explain the generalization of higher-order rogue waves in terms of rational solutions. We apply the contour line method to obtain the analytical formulas of the length and width of the first-order rogue wave of the complex mKdV and the nonlinear Schrödinger equations. In nonlinear optics, the higher-order rogue wave solutions found here will be very useful to generate high-power few-cycle optical pulses which will be applicable in the area of ultrashort pulse technology
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