55 research outputs found
-soliton solutions of the Fokas-Lenells equation for the plasma ion-cyclotron waves: Inverse scattering transform approach
We present a simple and constructive method to find -soliton solutions of
the equation suggested by Davydova and Lashkin to describe the dynamics of
nonlinear ion-cyclotron waves in a plasma and subsequently known (in a more
general form and as applied to nonlinear optics) as the Fokas-Lenells equation.
Using the classical inverse scattering transform approach, we find bright
-soliton solutions, rational -soliton solutions, and -soliton
solutions in the form of a mixture of exponential and rational functions.
Explicit breather solutions are presented as examples. Unlike purely algebraic
constructions of the Hirota or Darboux type, we also give a general expression
for arbitrary initial data decaying at infinity, which contains the
contribution of the continuous spectrum (radiation).Comment: arXiv admin note: text overlap with arXiv:2103.1009
A direct method of solution for the Fokas-Lenells derivative nonlinear Schr\"odinger equation: I. Bright soliton solutions
We develop a direct method of solution for finding the bright -soliton
solution of the Fokas-Lenells derivative nonlinear Schr\"odinger equation. The
construction of the solution is performed by means of a purely algebraic
procedure using an elementary theory of determinants and does not rely on the
inverse scattering transform method. We present two different expressions of
the solution both of which are expressed as a ratio of determinants. We then
investigate the properties of the solutions and find several new features.
Specifically, we derive the formula for the phase shift caused by the
collisions of bright solitons.Comment: To appear in J. Phys. A: Math. Theor. 45(2012) Ma
The Riemann-Hilbert approach for the integrable fractional Fokas--Lenells equation
In this paper, we propose a new integrable fractional Fokas--Lenells equation
by using the completeness of the squared eigenfunctions, dispersion relation,
and inverse scattering transform. To solve this equation, we employ the
Riemann-Hilbert approach. Specifically, we focus on the case of the
reflectionless potential with a simple pole for the zero boundary condition.
And we provide the fractional -soliton solution in determinant form.
Additionally, we prove the fractional one-soliton solution rigorously. Notably,
we demonstrate that as , the fractional -soliton solution can
be expressed as a linear combination of fractional single-soliton
solutions
The algebraic structure behind the derivative nonlinear Schroedinger equation
The Kaup-Newell (KN) hierarchy contains the derivative nonlinear Schr\"
odinger equation (DNLSE) amongst others interesting and important nonlinear
integrable equations. In this paper, a general higher grading affine algebraic
construction of integrable hierarchies is proposed and the KN hierarchy is
established in terms of a Kac-Moody algebra and principal
gradation. In this form, our spectral problem is linear in the spectral
parameter. The positive and negative flows are derived, showing that some
interesting physical models arise from the same algebraic structure. For
instance, the DNLSE is obtained as the second positive, while the Mikhailov
model as the first negative flows, respectively. The equivalence between the
latter and the massive Thirring model is explicitly demonstrated also. The
algebraic dressing method is employed to construct soliton solutions in a
systematic manner for all members of the hierarchy. Finally, the equivalence of
the spectral problem introduced in this paper with the usual one, which is
quadratic in the spectral parameter, is achieved by setting a particular
automorphism of the affine algebra, which maps the homogeneous into principal
gradation.Comment: references adde
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