1,447 research outputs found
Traveling waves of nonlinear Schr\"{o}dinger equation including higher order dispersions
The solitary wave solution and periodic solutions expressed in terms of
elliptic Jacobi's functions are obtained for the nonlinear Schr\"{o}dinger
equation governing the propagation of pulses in optical fibers including the
effects of second, third and fourth order dispersion. The approach is based on
the reduction of the generalized nonlinear Schr\"{o}dinger equation to an
ordinary nonlinear differential equation. The periodic solutions obtained form
one-parameter family which depend on an integration constant . The solitary
wave solution with shape is the limiting case of this family
with . The solutions obtained describe also a train of soliton-like pulses
with shape. It is shown that the bounded solutions arise only
for special domains of integration constant.Comment: We consider in this paper also the case with negative parameter
(defocusing nonlinearity
The Painleve Test and Reducibility to the Canonical Forms for Higher-Dimensional Soliton Equations with Variable-Coefficients
The general KdV equation (gKdV) derived by T. Chou is one of the famous (1+1)
dimensional soliton equations with variable coefficients. It is well-known that
the gKdV equation is integrable. In this paper a higher-dimensional gKdV
equation, which is integrable in the sense of the Painleve test, is presented.
A transformation that links this equation to the canonical form of the
Calogero-Bogoyavlenskii-Schiff equation is found. Furthermore, the form and
similar transformation for the higher-dimensional modified gKdV equation are
also obtained.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
Numerical study of blow-up and stability of line solitons for the Novikov-Veselov equation
We study numerically the evolution of perturbed Korteweg-de Vries solitons
and of well localized initial data by the Novikov-Veselov (NV) equation at
different levels of the "energy" parameter . We show that as , NV behaves, as expected, similarly to its formal limit, the
Kadomtsev-Petviashvili equation. However at intermediate regimes, i.e. when is not very large, more varied scenarios are possible, in particular,
blow-ups are observed. The mechanism of the blow-up is studied
On pattern structures of the N-soliton solution of the discrete KP equation over a finite field
The existence and properties of coherent pattern in the multisoliton
solutions of the dKP equation over a finite field is investigated. To that end,
starting with an algebro-geometric construction over a finite field, we derive
a "travelling wave" formula for -soliton solutions in a finite field.
However, despite it having a form similar to its analogue in the complex field
case, the finite field solutions produce patterns essentially different from
those of classical interacting solitons.Comment: 12 pages, 3 figure
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